mathematics
โข
3 likesโข106 views
This document provides information about circles and some circle theorems. It begins with definitions of basic circle terminology like radius, diameter, chord, arc, and sector. It then presents five circle theorems without proofs: 1) If two arcs are congruent, their chords are equal; 2) The perpendicular bisector of a chord passes through the center; 3) The line from the center to the midpoint of a chord is perpendicular to the chord; 4) There is only one circle passing through three non-collinear points; 5) The angle subtended by an arc at the center is double the angle at any other point on the remaining part of the circle. Sketch proofs are provided for theorems 2 through 5
Report
Share
Report
Share
Recommended
Circles
The document defines circles and related terms like radius, diameter, chord, arc, concentric circles, etc. It also presents some theorems regarding circles:
1. If two arcs of a circle are congruent, then the corresponding chords are equal.
2. If a chord is drawn perpendicular to another chord of a circle, then it bisects the chord.
3. The perpendicular bisector of a chord of a circle passes through the center of the circle.
4. There exists a unique circle passing through any three non-collinear points. The center of this circle lies at the intersection of the perpendicular bisectors of the chords formed by any two of the three points.
Circles
- A circle is a shape where all points are equidistant from the center point. Key terms include radius, diameter, chord, arc, sector, circumference, and tangent.
- The circumference of a circle is the distance around it and is calculated as 2ฯr. The area is calculated as ฯr^2.
- A tangent line to a circle is perpendicular to the radius drawn at the point of tangency. The lengths of tangents drawn from an external point to a circle are equal.
- Circles are abundant in nature and everyday life, appearing in architecture, vehicles, wheels, balls, and many sports. Without circles, common motions and activities would not be possible.
Circles for X class
This document discusses various concepts related to circles. It defines key terms like radius, diameter, circumference, chord, arc, secant, and tangent. It also presents two theorems - that the tangent at any point of a circle is perpendicular to the radius through that point, and that the lengths of two tangents drawn from an external point to a circle are equal. Examples of different types of intersections between circles and lines are provided. The document concludes by restating the key definitions and concepts covered.
Circle
The document provides a history and overview of circles. It discusses:
1) Euclid defined circles in 300 BC in his work "The Elements", establishing circles as fundamental objects in geometry. He defined a point and line and established early theorems about circles.
2) Key parts of a circle include the radius, diameter, circumference, chord, secant, tangent, arc, sector, and segment. The circumference is related to the diameter by pi.
3) Circles can be defined analytically using equations in the standard "center-radius" form of (x-h)2 + (y-k)2 = r2, where (h,k) are the coordinates of the
Circles class 9
1) The document discusses 10 theorems related to circles. Theorem 1 proves that equal chords of a circle subtend equal angles at the centre using congruent triangles.
2) Theorem 6 proves that the angle subtended by an arc at the centre is double the angle subtended by it at any point on the remaining part of the circle using angles on parallel lines.
3) Theorems 9 concludes that angles in the same segment of a circle are equal based on Theorem 6 and the definition of angles formed in a segment.
CH 10 CIRCLE PPT NCERT
1. The document defines key terms related to circles such as radius, diameter, center, chord, arc, and sector.
2. It presents 8 theorems about properties of circles and relationships between chords, arcs, angles, and points on a circle. The theorems prove properties such as equal chords subtend equal angles at the center, angles subtended by an arc are double the angle at any other point on the circle, and if a line segment subtends equal angles at two points, all four points lie on a circle.
3. Diagrams and formal proofs using triangle congruence or properties of angles are provided for each theorem.
Slideshare
This document contains 12 theorems regarding circles:
1. Equal chords of a circle subtend equal angles at the centre.
2. If the angles subtended by the chords of a circle at the centre are equal, then the chords are equal.
3. The perpendicular from the centre of a circle to a chord bisects the chord.
Circles 9th
This document defines key terms related to circles such as radius, diameter, chord, arc, and semicircle. It explains that a circle is a closed curve where all points are equidistant from the center. A radius is a line from the center to the edge, a diameter goes through the center and connects two points on the edge, and a chord connects any two points on the edge. An arc is part of the edge between two points, and a semicircle is half of the full circle. The document also describes properties of these terms like diameters being the longest chords and radii all being the same length.
Recommended
Circles
The document defines circles and related terms like radius, diameter, chord, arc, concentric circles, etc. It also presents some theorems regarding circles:
1. If two arcs of a circle are congruent, then the corresponding chords are equal.
2. If a chord is drawn perpendicular to another chord of a circle, then it bisects the chord.
3. The perpendicular bisector of a chord of a circle passes through the center of the circle.
4. There exists a unique circle passing through any three non-collinear points. The center of this circle lies at the intersection of the perpendicular bisectors of the chords formed by any two of the three points.
Circles
- A circle is a shape where all points are equidistant from the center point. Key terms include radius, diameter, chord, arc, sector, circumference, and tangent.
- The circumference of a circle is the distance around it and is calculated as 2ฯr. The area is calculated as ฯr^2.
- A tangent line to a circle is perpendicular to the radius drawn at the point of tangency. The lengths of tangents drawn from an external point to a circle are equal.
- Circles are abundant in nature and everyday life, appearing in architecture, vehicles, wheels, balls, and many sports. Without circles, common motions and activities would not be possible.
Circles for X class
This document discusses various concepts related to circles. It defines key terms like radius, diameter, circumference, chord, arc, secant, and tangent. It also presents two theorems - that the tangent at any point of a circle is perpendicular to the radius through that point, and that the lengths of two tangents drawn from an external point to a circle are equal. Examples of different types of intersections between circles and lines are provided. The document concludes by restating the key definitions and concepts covered.
Circle
The document provides a history and overview of circles. It discusses:
1) Euclid defined circles in 300 BC in his work "The Elements", establishing circles as fundamental objects in geometry. He defined a point and line and established early theorems about circles.
2) Key parts of a circle include the radius, diameter, circumference, chord, secant, tangent, arc, sector, and segment. The circumference is related to the diameter by pi.
3) Circles can be defined analytically using equations in the standard "center-radius" form of (x-h)2 + (y-k)2 = r2, where (h,k) are the coordinates of the
Circles class 9
1) The document discusses 10 theorems related to circles. Theorem 1 proves that equal chords of a circle subtend equal angles at the centre using congruent triangles.
2) Theorem 6 proves that the angle subtended by an arc at the centre is double the angle subtended by it at any point on the remaining part of the circle using angles on parallel lines.
3) Theorems 9 concludes that angles in the same segment of a circle are equal based on Theorem 6 and the definition of angles formed in a segment.
CH 10 CIRCLE PPT NCERT
1. The document defines key terms related to circles such as radius, diameter, center, chord, arc, and sector.
2. It presents 8 theorems about properties of circles and relationships between chords, arcs, angles, and points on a circle. The theorems prove properties such as equal chords subtend equal angles at the center, angles subtended by an arc are double the angle at any other point on the circle, and if a line segment subtends equal angles at two points, all four points lie on a circle.
3. Diagrams and formal proofs using triangle congruence or properties of angles are provided for each theorem.
Slideshare
This document contains 12 theorems regarding circles:
1. Equal chords of a circle subtend equal angles at the centre.
2. If the angles subtended by the chords of a circle at the centre are equal, then the chords are equal.
3. The perpendicular from the centre of a circle to a chord bisects the chord.
Circles 9th
This document defines key terms related to circles such as radius, diameter, chord, arc, and semicircle. It explains that a circle is a closed curve where all points are equidistant from the center. A radius is a line from the center to the edge, a diameter goes through the center and connects two points on the edge, and a chord connects any two points on the edge. An arc is part of the edge between two points, and a semicircle is half of the full circle. The document also describes properties of these terms like diameters being the longest chords and radii all being the same length.
Ix sumi
This document provides definitions and explanations of key concepts in circles:
- A circle is defined as the set of all points equidistant from a fixed point (the center).
- Key terms are defined like radius, diameter, circumference, chord, arc, sector, and tangent.
- The relationships between these terms are explained, such as diameter=2รradius.
- Examples are given to demonstrate determining if points lie inside, outside, or on the circle.
Cbse 10th circles
The document summarizes key definitions and properties related to circles and tangents to circles:
- It defines circles, radii, diameters, chords, secants, and tangents. It also defines tangent circles and common tangents.
- It states theorems about tangents, including that a tangent line is perpendicular to the radius at the point of tangency, and a line perpendicular to the radius at a point on the circle is a tangent.
- It provides examples demonstrating identifying special segments and lines related to circles, identifying common tangents, and using properties of tangents to solve problems.
Circles IX
Vaibhav Goel presented on circles and their properties. The presentation included definitions of key circle terms like radius, diameter, chord, and arc. It also proved several theorems: equal chords subtend equal angles at the center; a perpendicular from the center bisects a chord; there is one circle through three non-collinear points; equal chords are equidistant from the center; congruent arcs subtend equal angles; and the angle an arc subtends at the center is double that at any other point. The presentation concluded that angles in the same segment are equal and cyclic quadrilaterals have opposite angles summing to 180 degrees.
Circle - arc sector
This document discusses circles, arcs, sectors, and how to calculate their properties. It defines a circle as all points equidistant from a center point, and an arc as a portion of a circle's circumference. The length of an arc is calculated by taking the ratio of the arc's central angle measure to 360 degrees and multiplying by 2ฯr. Similarly, the area of a sector is calculated by taking the ratio of its central angle measure to 360 degrees and multiplying by ฯr^2. Several examples are provided to demonstrate calculating arc lengths and sector areas.
Circles-GEOMETRY
1. The document defines various terms related to circles such as radius, diameter, chord, arc, sector, segment, and circumference.
2. It states several properties of circles including that all radii of a circle are equal, the diameter of a circle is twice the radius, equal chords of a circle subtend equal angles at the centre, and there is one and only one circle passing through three non-collinear points.
3. Examples are provided to illustrate properties such as two arcs being congruent if their corresponding chords are equal, and the perpendicular drawn from the centre of a circle to a chord bisects the chord.
Circles
1. The document defines various terms related to circles like arc, chord, diameter, radius, and discusses 13 theorems related to circles.
2. The theorems discuss properties of circles like equal chords subtend equal angles, a perpendicular from the center bisects a chord, and the angle subtended by an arc at the center is double the angle at any other point on the circle.
3. The document concludes with a 16 point summary of the key topics covered related to defining circles and their geometric properties.
Tangents of circle
This presentation explains how to construct tangents to circles and the construction of incircles and circumcircles. It defines a tangent as a line that touches a circle at only one point, a chord as a line connecting two points on the circle, and a secant as a line intersecting the circle at two points. It provides steps for constructing tangents when circles intersect, touch internally or externally, or are concentric. It also gives the process for constructing a circumcircle around a triangle and the incircle inside a triangle.
Circle
1) The document defines and describes various terms related to circles such as radius, diameter, chord, arc, segment, and circumference.
2) A circle is a closed curve where all points are equidistant from the center. The radius is the line from the center to the edge, and the diameter passes through the center and joins two points on the edge.
3) Other terms defined are chord (a line through two points on the circle), arc (part of the circumference), segment (part of the region divided by a chord), and semicircle (half of a full circle).
Case study on circles
This document summarizes key concepts about circles. It defines circles and related terms like radius, diameter, chord, arc, and sector. It presents 8 theorems about angles subtended by chords and arcs, perpendiculars from the center to chords, circles through 3 points, equal chords and their distances from the center, and cyclic quadrilaterals. The concluding section summarizes that equal chords and arcs have corresponding relationships, angles in the same segment are equal, and properties of cyclic quadrilaterals. The document provides definitions, proofs, and conclusions about geometric properties of circles.
Angles in circle
The document discusses central angles and circumference angles in circles. A central angle is an angle inside a circle whose vertex is the center of the circle. A circumference angle is an angle whose vertex is located on the circumference of the circle. The central angle is twice the size of the circumference angle that subtends the same arc. Circumference angles that are subtended by the same arc are equal. Angles subtended by the diameter of a circle, forming a semicircle, are right angles. The document also discusses cyclic quadrilaterals, which have all four vertices lying on the circumference of a single circle, meaning the opposite angles of a cyclic quadrilateral are supplementary.
Circles
Circles are geometric figures whose points all lie the same fixed distance from a central point called the center. A circle is defined as the collection of all points in a plane that are equidistant from a fixed point. The line segment from the center to any point on the circle is called the radius, and the longest possible chord that passes through the center is the diameter, which has a length twice that of the radius. Circles have key properties like circumference, arcs, segments, and sectors that are defined in the document. Formulas are also provided relating circumference, diameter, radius, area of circles and circular segments.
11X1 T07 01 definitions & chord theorems
The document defines various terms related to circle geometry, including radius, diameter, chord, secant, tangent, arc, sector, segment, and more. It also presents several theorems regarding chords and arcs, such as a perpendicular from the circle's center bisects a chord, equal chords subtend equal angles at the center, and more. Diagrams with proofs are provided.
Introduction on Circle
1. A circle is defined as all points in a plane that are a fixed distance from a fixed center point. This fixed distance is called the radius.
2. Lines can intersect a circle in three ways: not at all, at one point (a tangent), or at two points (a secant). The longest secant that passes through the center is the diameter.
3. An arc is the portion of the circle cut off by a central angle. The measure of an arc is equal to the measure of its central angle.
maths ppt on circles
A circle is defined as all points in a plane that are equidistant from a fixed center point. The center point is called the center of the circle, and the fixed distance from the center is called the radius. The longest chord that can be drawn through the center is the diameter. If two chords of a circle are equal in length, then their distances from the center are also equal, as proven using the Side-Side-Side congruence rule for triangles.
10.1 tangents to circles
This document provides definitions and examples related to circles and tangents. It defines key terms like radius, diameter, chord, secant, and tangent. Examples demonstrate identifying these segments and determining if lines are tangent to circles. Theorems are presented about properties of tangents, such as tangents being perpendicular to radii and two tangents from the same exterior point being congruent. Proofs of theorems are also provided. Exercises apply these concepts, like using properties of tangents to find missing values.
Circles
1. The document defines key terms related to circles such as diameter, radius, chord, arc, and sector.
2. Several theorems about circles are presented, including that equal chords of a circle subtend equal angles at the center, and the perpendicular from the center of a circle to a chord bisects the chord.
3. The document summarizes that a circle can be defined as all points equidistant from a fixed point, and introduces various properties and relationships regarding angles, chords, and points on circles.
CHAPTER -10 CIRCLE 9TH CLASS NCERT
The document defines key terms related to circles such as radius, diameter, chord, arc, and sector. It discusses properties of circles including: angles subtended by chords; perpendiculars from the center to chords bisect chords; there is one unique circle through three non-collinear points; equal chords are equidistant from the center; congruent arcs subtend equal angles; and the sums of opposite angles in a cyclic quadrilateral are 180 degrees. The document concludes by summarizing key properties of circles.
10.6 circles and arcs
The document defines key terms related to circles such as central angle, arc, semicircle, minor arc, and major arc. It provides examples of identifying these elements in circles and finding the measures of central angles and arcs. The document concludes with homework problems involving calculating measures of central angles and arcs in various circle graphs and diagrams.
Circle : Basic concept
This document defines key terms related to circles:
- A circle consists of all points equidistant from a fixed center point.
- The radius is the fixed distance from the center point to any point on the circle.
- A chord connects any two points on a circle. A diameter passes through the center and divides the circle into two equal halves.
- The standard equation of a circle with center (h, k) and radius r is (x - h)2 + (y - k)2 = r2.
Circle and its parts
This document defines and describes various parts of a circle including the radius, diameter, chord, arc, secant, and tangent. It explains that a circle is a closed curve where all points are equidistant from the center. A radius is a line from the center to the edge, a diameter connects two points on the edge passing through the center, and a chord connects any two edge points. An arc is part of the edge between two points, and a semicircle is half of a full circle. Secants and tangents are lines that intersect the circle at one or more points.
CIRCLE math 10 Second Quarter PowerPoint
The document discusses circles and related concepts:
- It defines a circle and its key elements like radius, diameter, circumference, chord, arc, and sector.
- It explains applications of circles in daily life and important terms like tangent, secant, and their properties regarding circles.
- It provides examples of problems involving calculating radii based on tangent lengths, finding angles between tangents, and proving properties of tangents drawn to circles.
Class 10th Chapter 4 Circles.pptx
The document discusses various properties of circles and tangents to circles. It defines a circle and discusses what happens when a circle and line intersect in a plane. It defines a tangent as a line that touches a circle at only one point and proves that the tangent is perpendicular to the radius at the point of contact. It discusses the number of tangents that can be drawn from a point and proves that tangents from an external point are equal in length. Examples and exercises are provided to illustrate the concepts.
More Related Content
What's hot
Ix sumi
This document provides definitions and explanations of key concepts in circles:
- A circle is defined as the set of all points equidistant from a fixed point (the center).
- Key terms are defined like radius, diameter, circumference, chord, arc, sector, and tangent.
- The relationships between these terms are explained, such as diameter=2รradius.
- Examples are given to demonstrate determining if points lie inside, outside, or on the circle.
Cbse 10th circles
The document summarizes key definitions and properties related to circles and tangents to circles:
- It defines circles, radii, diameters, chords, secants, and tangents. It also defines tangent circles and common tangents.
- It states theorems about tangents, including that a tangent line is perpendicular to the radius at the point of tangency, and a line perpendicular to the radius at a point on the circle is a tangent.
- It provides examples demonstrating identifying special segments and lines related to circles, identifying common tangents, and using properties of tangents to solve problems.
Circles IX
Vaibhav Goel presented on circles and their properties. The presentation included definitions of key circle terms like radius, diameter, chord, and arc. It also proved several theorems: equal chords subtend equal angles at the center; a perpendicular from the center bisects a chord; there is one circle through three non-collinear points; equal chords are equidistant from the center; congruent arcs subtend equal angles; and the angle an arc subtends at the center is double that at any other point. The presentation concluded that angles in the same segment are equal and cyclic quadrilaterals have opposite angles summing to 180 degrees.
Circle - arc sector
This document discusses circles, arcs, sectors, and how to calculate their properties. It defines a circle as all points equidistant from a center point, and an arc as a portion of a circle's circumference. The length of an arc is calculated by taking the ratio of the arc's central angle measure to 360 degrees and multiplying by 2ฯr. Similarly, the area of a sector is calculated by taking the ratio of its central angle measure to 360 degrees and multiplying by ฯr^2. Several examples are provided to demonstrate calculating arc lengths and sector areas.
Circles-GEOMETRY
1. The document defines various terms related to circles such as radius, diameter, chord, arc, sector, segment, and circumference.
2. It states several properties of circles including that all radii of a circle are equal, the diameter of a circle is twice the radius, equal chords of a circle subtend equal angles at the centre, and there is one and only one circle passing through three non-collinear points.
3. Examples are provided to illustrate properties such as two arcs being congruent if their corresponding chords are equal, and the perpendicular drawn from the centre of a circle to a chord bisects the chord.
Circles
1. The document defines various terms related to circles like arc, chord, diameter, radius, and discusses 13 theorems related to circles.
2. The theorems discuss properties of circles like equal chords subtend equal angles, a perpendicular from the center bisects a chord, and the angle subtended by an arc at the center is double the angle at any other point on the circle.
3. The document concludes with a 16 point summary of the key topics covered related to defining circles and their geometric properties.
Tangents of circle
This presentation explains how to construct tangents to circles and the construction of incircles and circumcircles. It defines a tangent as a line that touches a circle at only one point, a chord as a line connecting two points on the circle, and a secant as a line intersecting the circle at two points. It provides steps for constructing tangents when circles intersect, touch internally or externally, or are concentric. It also gives the process for constructing a circumcircle around a triangle and the incircle inside a triangle.
Circle
1) The document defines and describes various terms related to circles such as radius, diameter, chord, arc, segment, and circumference.
2) A circle is a closed curve where all points are equidistant from the center. The radius is the line from the center to the edge, and the diameter passes through the center and joins two points on the edge.
3) Other terms defined are chord (a line through two points on the circle), arc (part of the circumference), segment (part of the region divided by a chord), and semicircle (half of a full circle).
Case study on circles
This document summarizes key concepts about circles. It defines circles and related terms like radius, diameter, chord, arc, and sector. It presents 8 theorems about angles subtended by chords and arcs, perpendiculars from the center to chords, circles through 3 points, equal chords and their distances from the center, and cyclic quadrilaterals. The concluding section summarizes that equal chords and arcs have corresponding relationships, angles in the same segment are equal, and properties of cyclic quadrilaterals. The document provides definitions, proofs, and conclusions about geometric properties of circles.
Angles in circle
The document discusses central angles and circumference angles in circles. A central angle is an angle inside a circle whose vertex is the center of the circle. A circumference angle is an angle whose vertex is located on the circumference of the circle. The central angle is twice the size of the circumference angle that subtends the same arc. Circumference angles that are subtended by the same arc are equal. Angles subtended by the diameter of a circle, forming a semicircle, are right angles. The document also discusses cyclic quadrilaterals, which have all four vertices lying on the circumference of a single circle, meaning the opposite angles of a cyclic quadrilateral are supplementary.
Circles
Circles are geometric figures whose points all lie the same fixed distance from a central point called the center. A circle is defined as the collection of all points in a plane that are equidistant from a fixed point. The line segment from the center to any point on the circle is called the radius, and the longest possible chord that passes through the center is the diameter, which has a length twice that of the radius. Circles have key properties like circumference, arcs, segments, and sectors that are defined in the document. Formulas are also provided relating circumference, diameter, radius, area of circles and circular segments.
11X1 T07 01 definitions & chord theorems
The document defines various terms related to circle geometry, including radius, diameter, chord, secant, tangent, arc, sector, segment, and more. It also presents several theorems regarding chords and arcs, such as a perpendicular from the circle's center bisects a chord, equal chords subtend equal angles at the center, and more. Diagrams with proofs are provided.
Introduction on Circle
1. A circle is defined as all points in a plane that are a fixed distance from a fixed center point. This fixed distance is called the radius.
2. Lines can intersect a circle in three ways: not at all, at one point (a tangent), or at two points (a secant). The longest secant that passes through the center is the diameter.
3. An arc is the portion of the circle cut off by a central angle. The measure of an arc is equal to the measure of its central angle.
maths ppt on circles
A circle is defined as all points in a plane that are equidistant from a fixed center point. The center point is called the center of the circle, and the fixed distance from the center is called the radius. The longest chord that can be drawn through the center is the diameter. If two chords of a circle are equal in length, then their distances from the center are also equal, as proven using the Side-Side-Side congruence rule for triangles.
10.1 tangents to circles
This document provides definitions and examples related to circles and tangents. It defines key terms like radius, diameter, chord, secant, and tangent. Examples demonstrate identifying these segments and determining if lines are tangent to circles. Theorems are presented about properties of tangents, such as tangents being perpendicular to radii and two tangents from the same exterior point being congruent. Proofs of theorems are also provided. Exercises apply these concepts, like using properties of tangents to find missing values.
Circles
1. The document defines key terms related to circles such as diameter, radius, chord, arc, and sector.
2. Several theorems about circles are presented, including that equal chords of a circle subtend equal angles at the center, and the perpendicular from the center of a circle to a chord bisects the chord.
3. The document summarizes that a circle can be defined as all points equidistant from a fixed point, and introduces various properties and relationships regarding angles, chords, and points on circles.
CHAPTER -10 CIRCLE 9TH CLASS NCERT
The document defines key terms related to circles such as radius, diameter, chord, arc, and sector. It discusses properties of circles including: angles subtended by chords; perpendiculars from the center to chords bisect chords; there is one unique circle through three non-collinear points; equal chords are equidistant from the center; congruent arcs subtend equal angles; and the sums of opposite angles in a cyclic quadrilateral are 180 degrees. The document concludes by summarizing key properties of circles.
10.6 circles and arcs
The document defines key terms related to circles such as central angle, arc, semicircle, minor arc, and major arc. It provides examples of identifying these elements in circles and finding the measures of central angles and arcs. The document concludes with homework problems involving calculating measures of central angles and arcs in various circle graphs and diagrams.
Circle : Basic concept
This document defines key terms related to circles:
- A circle consists of all points equidistant from a fixed center point.
- The radius is the fixed distance from the center point to any point on the circle.
- A chord connects any two points on a circle. A diameter passes through the center and divides the circle into two equal halves.
- The standard equation of a circle with center (h, k) and radius r is (x - h)2 + (y - k)2 = r2.
Circle and its parts
This document defines and describes various parts of a circle including the radius, diameter, chord, arc, secant, and tangent. It explains that a circle is a closed curve where all points are equidistant from the center. A radius is a line from the center to the edge, a diameter connects two points on the edge passing through the center, and a chord connects any two edge points. An arc is part of the edge between two points, and a semicircle is half of a full circle. Secants and tangents are lines that intersect the circle at one or more points.
What's hot (20)
Similar to mathematics
CIRCLE math 10 Second Quarter PowerPoint
The document discusses circles and related concepts:
- It defines a circle and its key elements like radius, diameter, circumference, chord, arc, and sector.
- It explains applications of circles in daily life and important terms like tangent, secant, and their properties regarding circles.
- It provides examples of problems involving calculating radii based on tangent lengths, finding angles between tangents, and proving properties of tangents drawn to circles.
Class 10th Chapter 4 Circles.pptx
The document discusses various properties of circles and tangents to circles. It defines a circle and discusses what happens when a circle and line intersect in a plane. It defines a tangent as a line that touches a circle at only one point and proves that the tangent is perpendicular to the radius at the point of contact. It discusses the number of tangents that can be drawn from a point and proves that tangents from an external point are equal in length. Examples and exercises are provided to illustrate the concepts.
class 10 circles
- A circle is the set of all points in a plane that are equidistant from a fixed point called the center.
- Important terms related to circles include chord, diameter, arc, sector, minor/major segments.
- A tangent touches the circle at one point, a secant intersects at two points, and there can be at most two parallel tangents for a given secant.
- The tangent radius theorem states that the tangent is perpendicular to the radius at the point of contact. The equal tangent lengths theorem says tangents from an external point are equal in length.
Circles
This document provides definitions and theorems related to circles. It defines circle terms like radius, diameter, chord, arc, tangent, and sector. It proves theorems such as a perpendicular from the center of a circle to a chord bisects the chord; equal chords subtend equal angles; and the angle subtended by an arc at the center is double the angle at any other point on the circle. It also proves properties of tangents, like tangents from an external point being equal lengths.
Circle and sphere
A circle is defined as the set of all points in a plane that are equidistant from a fixed point called the center. A sphere is a three-dimensional shape where all points on its surface are equidistant from its center point. Both circles and spheres have perfect symmetry around their centers, with all points a distance r from the center forming the shape. While a circle is a two-dimensional figure defined by its radius, a sphere is a three-dimensional object with volume in addition to a surface area, defined as 4ฯr^2 and (4/3)ฯr^3 respectively.
circles-131126094958-phpapp01.pdf
Hey to go back to m Al and resume to this email and resume y to the measure of interest in and out to go back and resume to this post to you y y y y y y y y y t y y y y y y to y y to y to the measure and resume for your help to learn to learn to learn to learn y y u yum install the world of SRP RQS PRQ PQS to go back to m Al to be a t shirt ๐ and go for the world ๐ to be a t y to the to do the world i the world ๐ and pink y to y y u can you must i and white ๐ค to be a I will be in and pink rod is Shreya aur Priyanshi and resume tow bar ๐บ to be a t y t y to the to go in I will be in the measure to be a t y t y to the world ๐ to go to go with the world of interest in the morning ๐๐๐๐๐๐๐๐
๐
๐
to be a good mood to this email โ๏ธโ๏ธ๐จ to go to go to go with you and your help in the morning ๐ to be in and pink y to y y u yum ๐๐๐๐๐๐๐๐๐ to go to go to go to this email ๐จ๐จ๐จ to go to be a good ๐๐๐ to go to go with you and pink and resume for your I I am in the measure to go to go I I I I I I I am in the measure of my friends go for it to go with you to get scholarship to this video ๐ธ๐ธ๐ธ๐ธ๐ธ to go to reel featuring David and resume for two ๐๐๐ to go I have to check the to you y u sir to be in touch and white ๐ปโโ๏ธ๐ปโโ๏ธ to be of any kind and white ๐ปโโ๏ธ๐ปโโ๏ธ to go with you and your help and white ๐ปโโ๏ธ๐ปโโ๏ธ๐ปโโ๏ธ๐ปโโ๏ธ๐ปโโ๏ธ๐ปโโ๏ธ to be in and out go back to you and pink y u can you send the measure of my to be a t to this video ๐ธ๐ธ๐ธ to go with you and pink y u sir to send the measure to be of interest in our country and white ๐ปโโ๏ธ๐ปโโ๏ธ๐ปโโ๏ธ๐ปโโ๏ธ to go to you y y to y u yum ๐๐ to go to m go to sleep ๐ด๐ด๐ด๐ด๐ด to be in touch with you must i to be of any other information to you must have been working in a little bit more time to get scholarship for two ๐๐๐๐ to be a little t to a t shirt ๐ฝ๐ฝ๐ฝ๐ฝ to go to go to go to go to school ๐ซ๐ซ๐ซ to be a little t y t to go to learn y u can be of SRP to be in the world ๐๐ to go to learn more about the measure to be of any action taken to be a little bit of interest for you must i to go with a t to go with the same shoes like to know about this video is currently a good ๐๐๐๐๐๐๐ to be of SRP to be in and out go talking with a t shirt ๐ฝ๐ฝ๐ฝ to go to go to go with a good ๐๐๐ to go to go to go to go with you must i to go with the measure of interest in the morning ๐
๐
to go with a little more than a t y y y y y y y y y y y y y u sir to be of any action in and out go to sleep now and white striped shirt and they are in a t y y to go to go with the same to u dear and white striped shirt and pink y to go with a good ๐๐๐๐๐ to go to go to go to go to go to go to go to go to go to go to go to go to go to go to go to go to go to go with the same to you must have been a little yyyyyyy to go to go to go to go to go to go to go to go to go to go to go to go to go to go to sleep ๐ค๐ด๐ด๐ด and resume for your help to learn to learn by you and your family a very happy birthday dear fr
2.4. l1. area of circle
The document discusses the area of a circle. It defines a circle as a locus of points whose distance from a fixed point is always constant. The fixed point is called the center. It provides key terms like radius, diameter, chord, arc, and circumference. It then derives the formula for the area of a circle as ฯr^2, where r is the radius. Some examples are provided to demonstrate calculating area given radius or circumference. Finally, applications of circles in daily life like wheels and wells are briefly described.
Areas related to Circles - class 10 maths
This a ppt which is based on chapter circles of class 10 maths it is a very good ppt which will definitely enhance your knowledge . it will also clear all concepts and doubts about this chapter and its topics
mathemaics 10 lesson about cicles. its part
grade 10 lesson about circle. mathematocs lesson about circles. parts of the circles and the corresponding measurements.grade 10 lesson about circle. mathematocs lesson about circles. parts of the circles and the corresponding measurements. grade 10 lesson about circle. mathematocs lesson about circles. parts of the circles and the corresponding measurements. grade 10 lesson about circle. mathematocs lesson about circles. parts of the circles and the corresponding measurements. grade 10 lesson about circle. mathematocs lesson about circles. parts of the circles and the corresponding measurements. grade 10 lesson about circle. mathematocs lesson about circles. parts of the circles and the corresponding measurements. grade 10 lesson about circle. mathematocs lesson about circles. parts of the circles and the corresponding measurements. grade 10 lesson about circle. mathematocs lesson about circles. parts of the circles and the corresponding measurements. grade 10 lesson about circle. mathematocs lesson about circles. parts of the circles and the corresponding measurements. grade 10 lesson about circle. mathematocs lesson about circles. parts of the circles and the corresponding measurements. grade 10 lesson about circle. mathematocs lesson about circles. parts of the circles and the corresponding measurements. grade 10 lesson about circle. mathematocs lesson about circles. parts of the circles and the corresponding measurements. grade 10 lesson about circle. mathematocs lesson about circles. parts of the circles and the corresponding measurements. grade 10 lesson about circle. mathematocs lesson about circles. parts of the circles and the corresponding measurements. grade 10 lesson about circle. mathematocs lesson about circles. parts of the circles and the corresponding measurements. grade 10 lesson about circle. mathematocs lesson about circles. parts of the circles and the corresponding measurements. grade 10 lesson about circle. mathematocs lesson about circles. parts of the circles and the corresponding measurements. grade 10 lesson about circle. mathematocs lesson about circles. parts of the circles and the corresponding measurements. grade 10 lesson about circle. mathematocs lesson about circles. parts of the circles and the corresponding measurements. grade 10 lesson about circle. mathematocs lesson about circles. parts of the circles and the corresponding measurements. grade 10 lesson about circle. mathematocs lesson about circles. parts of the circles and the corresponding measurements. grade 10 lesson about circle. mathematocs lesson about circles. parts of the circles and the corresponding measurements. grade 10 lesson about circle. mathematocs lesson about circles. parts of the circles and the corresponding measurements. grade 10 lesson about circle. mathematocs lesson about circles. parts of the circles and the corresponding measurements. grade 10 lesson about circle. mathematocs lesson about circles. parts of the circles and the corr
Circle for class 10 by G R Ahmed,TGT(Maths) at K V Khanapara
1) The document discusses various properties related to circles, chords, tangents, and radii. It defines what a chord, secant, tangent and radius are with reference to a circle.
2) It proves several theorems about tangents including that the angle between a tangent and radius at the point of contact is 90 degrees and the angle between a tangent and a chord through the point of contact is equal to the angle subtended by the chord in the alternate segment.
3) It also demonstrates how to draw a tangent and secant to a circle parallel to a given line and proves that tangents drawn at the ends of a diameter are parallel.
1. circles and theorems - introduction
The document discusses theorems related to tangents of circles. It begins by revising circle concepts such as defining a circle, radius, diameter, chord, arc, sector, and tangent. It then proves two theorems: 1) the tangent at any point of a circle is perpendicular to the radius through the point of contact, and 2) the lengths of tangents drawn from an external point to a circle are equal. Diagrams and step-by-step proofs are provided for each theorem.
Areas related to circle
1) A circle is defined as the set of all points in a plane that are equidistant from a fixed point called the center.
2) Key terms related to circles include radius, diameter, chord, arc, sector, and segment. The radius connects the center to any point on the circle, while the diameter connects two points on the circle and passes through the center.
3) The area of a circle is calculated as ฯr^2, where r is the radius. The area of a sector of a circle is calculated based on the central angle in radians that the sector spans.
CIRCLES
This document discusses various theorems related to circles:
1. Equal chords of a circle subtend equal angles at the centre.
2. If the angles subtended by the chords of a circle at the centre are equal, then the chords are equal.
3. The perpendicular from the centre of a circle to a chord bisects the chord.
Curves2
The document discusses various types of engineering curves including involutes, cycloids, spirals, and helices. It provides definitions for involutes, cycloids, epicycloids, hypotrochoids, spirals, and helices. Examples are given on how to draw involutes of circles, squares, and triangles. Methods for drawing tangents and normals to involutes, cycloids, and epicycloids are also described. Problems include drawing loci for points on circles rolling along straight or curved paths to form different types of cycloids.
Area of the circle
This document provides information about calculating the area of a circle. It defines a circle and key terms like radius, diameter, chord, and circumference. It then derives the formula for the area of a circle as A = ฯr^2 by dividing a circle into sectors and arranging them to form a rectangle. Some examples are provided to demonstrate calculating the area given the radius or circumference. The document concludes by stating the area formula can be used to find the area of circular objects and shapes in daily life.
Maths Circle PPT Class10
This document summarizes key terms and theorems related to circles:
1. It defines circles and related terms like radius, diameter, chord, arc, and sector.
2. It describes theorems like equal chords subtend equal angles at the center, and conversely if angles are equal then chords are equal.
3. Other concepts covered include perpendiculars from the center bisect chords, congruent arcs subtend equal angles, and cyclic quadrilaterals have opposite angles summing to 180 degrees.
Maths Art Integrated Activity 2022-23.pptx
1. The document discusses various terms related to circles such as radius, diameter, chord, arc, segment, and sector. It defines each term and provides examples.
2. Formulas for calculating the circumference and area of a circle are presented. Circumference is defined as 2ฯr and area is defined as ฯr^2.
3. Methods for calculating the area of a sector and segment of a circle are described. The area of a sector is calculated based on the central angle and area of the whole circle. The area of a segment is not explicitly defined.
3 circle 1
Learn about the properties of tangents, chords and arcs of the circle. Learn to find measure of the inscribed angle and the property of cyclic quadrilateral
circles-.pptx
The document defines and describes the key parts of a circle, including the radius, diameter, chord, arc, central angle, inscribed angle, semicircle, and sector. It provides examples and diagrams to illustrate each part. The document also covers theorems about central angles, arcs and chords in circles, as well as the area of sectors and segments of a circle.
Similar to mathematics (20)
Circle for class 10 by G R Ahmed,TGT(Maths) at K V Khanapara
Circle for class 10 by G R Ahmed,TGT(Maths) at K V Khanapara
ย
Recently uploaded
The History of Stoke Newington Street Names
Presented at the Stoke Newington Literary Festival on 9th June 2024
www.StokeNewingtonHistory.com
A Independรชncia da Amรฉrica Espanhola LAPBOOK.pdf
Lapbook sobre independรชncia da Amรฉrica Espanhola.
PCOS corelations and management through Ayurveda.
This presentation includes basic of PCOS their pathology and treatment and also Ayurveda correlation of PCOS and Ayurvedic line of treatment mentioned in classics.
The Diamonds of 2023-2024 in the IGRA collection
A review of the growth of the Israel Genealogy Research Association Database Collection for the last 12 months. Our collection is now passed the 3 million mark and still growing. See which archives have contributed the most. See the different types of records we have, and which years have had records added. You can also see what we have for the future.
LAND USE LAND COVER AND NDVI OF MIRZAPUR DISTRICT, UP
This Dissertation explores the particular circumstances of Mirzapur, a region located in the
core of India. Mirzapur, with its varied terrains and abundant biodiversity, offers an optimal
environment for investigating the changes in vegetation cover dynamics. Our study utilizes
advanced technologies such as GIS (Geographic Information Systems) and Remote sensing to
analyze the transformations that have taken place over the course of a decade.
The complex relationship between human activities and the environment has been the focus
of extensive research and worry. As the global community grapples with swift urbanization,
population expansion, and economic progress, the effects on natural ecosystems are becoming
more evident. A crucial element of this impact is the alteration of vegetation cover, which plays a
significant role in maintaining the ecological equilibrium of our planet.Land serves as the foundation for all human activities and provides the necessary materials for
these activities. As the most crucial natural resource, its utilization by humans results in different
'Land uses,' which are determined by both human activities and the physical characteristics of the
land.
The utilization of land is impacted by human needs and environmental factors. In countries
like India, rapid population growth and the emphasis on extensive resource exploitation can lead
to significant land degradation, adversely affecting the region's land cover.
Therefore, human intervention has significantly influenced land use patterns over many
centuries, evolving its structure over time and space. In the present era, these changes have
accelerated due to factors such as agriculture and urbanization. Information regarding land use and
cover is essential for various planning and management tasks related to the Earth's surface,
providing crucial environmental data for scientific, resource management, policy purposes, and
diverse human activities.
Accurate understanding of land use and cover is imperative for the development planning
of any area. Consequently, a wide range of professionals, including earth system scientists, land
and water managers, and urban planners, are interested in obtaining data on land use and cover
changes, conversion trends, and other related patterns. The spatial dimensions of land use and
cover support policymakers and scientists in making well-informed decisions, as alterations in
these patterns indicate shifts in economic and social conditions. Monitoring such changes with the
help of Advanced technologies like Remote Sensing and Geographic Information Systems is
crucial for coordinated efforts across different administrative levels. Advanced technologies like
Remote Sensing and Geographic Information Systems
9
Changes in vegetation cover refer to variations in the distribution, composition, and overall
structure of plant communities across different temporal and spatial scales. These changes can
occur natural.
How to Setup Warehouse & Location in Odoo 17 Inventory
In this slide, we'll explore how to set up warehouses and locations in Odoo 17 Inventory. This will help us manage our stock effectively, track inventory levels, and streamline warehouse operations.
How to Make a Field Mandatory in Odoo 17
In Odoo, making a field required can be done through both Python code and XML views. When you set the required attribute to True in Python code, it makes the field required across all views where it's used. Conversely, when you set the required attribute in XML views, it makes the field required only in the context of that particular view.
The simplified electron and muon model, Oscillating Spacetime: The Foundation...
Discover the Simplified Electron and Muon Model: A New Wave-Based Approach to Understanding Particles delves into a groundbreaking theory that presents electrons and muons as rotating soliton waves within oscillating spacetime. Geared towards students, researchers, and science buffs, this book breaks down complex ideas into simple explanations. It covers topics such as electron waves, temporal dynamics, and the implications of this model on particle physics. With clear illustrations and easy-to-follow explanations, readers will gain a new outlook on the universe's fundamental nature.
ANATOMY AND BIOMECHANICS OF HIP JOINT.pdf
it describes the bony anatomy including the femoral head , acetabulum, labrum . also discusses the capsule , ligaments . muscle that act on the hip joint and the range of motion are outlined. factors affecting hip joint stability and weight transmission through the joint are summarized.
Advanced Java[Extra Concepts, Not Difficult].docx
This is part 2 of my Java Learning Journey. This contains Hashing, ArrayList, LinkedList, Date and Time Classes, Calendar Class and more.
Pollock and Snow "DEIA in the Scholarly Landscape, Session One: Setting Expec...
Pollock and Snow "DEIA in the Scholarly Landscape, Session One: Setting Expec...National Information Standards Organization (NISO)
This presentation was provided by Steph Pollock of The American Psychological Associationโs Journals Program, and Damita Snow, of The American Society of Civil Engineers (ASCE), for the initial session of NISO's 2024 Training Series "DEIA in the Scholarly Landscape." Session One: 'Setting Expectations: a DEIA Primer,' was held June 6, 2024.ย
How to Add Chatter in the odoo 17 ERP Module
In Odoo, the chatter is like a chat tool that helps you work together on records. You can leave notes and track things, making it easier to talk with your team and partners. Inside chatter, all communication history, activity, and changes will be displayed.
clinical examination of hip joint (1).pdf
described clinical examination all orthopeadic conditions .
Walmart Business+ and Spark Good for Nonprofits.pdf
"Learn about all the ways Walmart supports nonprofit organizations.
You will hear from Liz Willett, the Head of Nonprofits, and hear about what Walmart is doing to help nonprofits, including Walmart Business and Spark Good. Walmart Business+ is a new offer for nonprofits that offers discounts and also streamlines nonprofits order and expense tracking, saving time and money.
The webinar may also give some examples on how nonprofits can best leverage Walmart Business+.
The event will cover the following::
Walmart Business + (https://business.walmart.com/plus) is a new shopping experience for nonprofits, schools, and local business customers that connects an exclusive online shopping experience to stores. Benefits include free delivery and shipping, a 'Spend Analyticsโ feature, special discounts, deals and tax-exempt shopping.
Special TechSoup offer for a free 180 days membership, and up to $150 in discounts on eligible orders.
Spark Good (walmart.com/sparkgood) is a charitable platform that enables nonprofits to receive donations directly from customers and associates.
Answers about how you can do more with Walmart!"
Chapter 4 - Islamic Financial Institutions in Malaysia.pptx
Chapter 4 - Islamic Financial Institutions in Malaysia.pptxMohd Adib Abd Muin, Senior Lecturer at Universiti Utara Malaysia
This slide is special for master students (MIBS & MIFB) in UUM. Also useful for readers who are interested in the topic of contemporary Islamic banking.
ย
Recently uploaded (20)
A Independรชncia da Amรฉrica Espanhola LAPBOOK.pdf
A Independรชncia da Amรฉrica Espanhola LAPBOOK.pdf
ย
Liberal Approach to the Study of Indian Politics.pdf
Liberal Approach to the Study of Indian Politics.pdf
ย
LAND USE LAND COVER AND NDVI OF MIRZAPUR DISTRICT, UP
LAND USE LAND COVER AND NDVI OF MIRZAPUR DISTRICT, UP
ย
How to Setup Warehouse & Location in Odoo 17 Inventory
How to Setup Warehouse & Location in Odoo 17 Inventory
ย
The simplified electron and muon model, Oscillating Spacetime: The Foundation...
The simplified electron and muon model, Oscillating Spacetime: The Foundation...
ย
Pollock and Snow "DEIA in the Scholarly Landscape, Session One: Setting Expec...
Pollock and Snow "DEIA in the Scholarly Landscape, Session One: Setting Expec...
ย
Walmart Business+ and Spark Good for Nonprofits.pdf
Walmart Business+ and Spark Good for Nonprofits.pdf
ย
Chapter 4 - Islamic Financial Institutions in Malaysia.pptx
Chapter 4 - Islamic Financial Institutions in Malaysia.pptx
ย
mathematics
- 3. CONTENTS ๏Introduction ๏Arc of circle ๏ chord diameter of circle ๏Area of circle ๏Circles in real life application
- 4. INTRODUCTION โข CIRCLE โข A circle is a collection of all points in a plane that arc at a given fixed point in th The distance between any of the points and the centre is called the radius. It can also be defined as the locus of a point equidistant from a fixed point. e plane โข . A circle is a simple closed curve which divides the plane into two regions: an interior and an exterior. In everyday use, the term "circle" may be used interchangeably to refer to either the boundary of the figure, or to the whole figure including its interior; in strict technical usage, the circle is the former and the latter is called a disk.
- 5. Arc of circle A continuous piece of circle is called arc of a circle Minor arc A collection of those points of the circle That lie on and also inside a central angle Major arc major arc of a circle is the collection of points of the circle that lie on and outside a central angle
- 6. Chord, diameter,semicircle A line segment joining any two points on a circle is called a chord of circle A chord passing through centre of circle is known as diameter of circle A diameter of circle divides divides it into two equal parts which are arcs each of this two arcs is called semicircle
- 7. Area of circle โข The ratio of a circle's circumference to its diameter is ฯ (pi), an irrational constant approximately equal to 3.141592654. Thus the length of the circumference C is related to the radius r and diameter d by:
- 8. Circles in real life application This is a clock. It has different shapes. Your clock can be a square- shape. But have you notice whatโs inside? I have once open a destroyed one. It has a gear inside. Do you know what a gear is? What is its shape? Yes itโs like a circle. Have you seen a gear that has a shape other than circle? Difinitely, you can not see a gear that has a shape other than circle
- 10. THEOREM 1 If two arcs of circle are congruent then corresponding chords are equal
- 11. Sol: โข Given, โข Arc pq =arc ab and also congruent โข TO PROVE :- PQ=AB โข PROOF โข In OPQ andOAB โข OP=OQ=OโR=OโB (Equal radii of two circles ) โข POQ= ROโS(ARC PQ =arc AB =>m{arc PQ }=m {AB} โข So by sas criterion of congruence we have POQ CONGRUENT โข AOB SO PQ=AB {C.P.C.T} โข HENCE PROVED
- 12. THEOREM 2 The perpendicular from center of circle bisect the chord
- 13. SOI: โข Given โข A chord PQ of a circle โข Cons = join OP to OQ โข Proof in triangle PLO and QLO we have โข OP =OQ=r {radii of same circle} โข OL=OL (commom side ) โข OLP= OLQ (each angle right angle) โข So by RHS criterion of congruence we have PLO= QLO
- 14. THE LINE JOINING THE CENTRE OF CIRCLE TO THE MID POINT OF CHORD IS PERPEN DI CULAR TO CHORD
- 15. SOL:- โข Given:- A chord PQ of a circle with mid point m โข Cons:-join OP to OQ โข PROOF :- in triangle OPM and OQM โข OP=OQ =r (radii of same circle) โข PM =MQ (M is mid point ) โข OM=OM (COMMON SIDE} โข OPM= OQM โข OMP= OMP (C.P.C.T) โข OMP +OMQ=180 (LINEAR PAIR ) โข 2 OMP =180 โข OMP = 90 โข OMP=OMQ =90 โข OM PQ
- 16. THERE ARE ONE AND ONLY ONE CIRCLE PASSING THROUGH THREE NON CO LINEAR POINTS J K
- 17. SOL : โข GIVEN ,three non collinear point P,Q,R AND O IS CENTRE OF CIRCLE TO PROVE THERE IS ONLY ONE CIRCLE PASS THROUGH THREE NON COLLINEAR POINT CONS= JOIN P TO Q TO R DRAW PERPENDICULAR BISECTOR AL AND BM OF PQ AND RQ ALSO WHERE BOTH PERPENDICULAR BISECTOR MEET NAME IT O;JOIN OR,OQ,OP PROOF = IN TRIANGLE POJ AND QOJ OJ=OJ (COMMON SIDE) OJP=OJQ (EACH 90 ) PJ=JQ (PERPENDICULAR BISECTOR BISECT SIDE ) SO TRIANGLE OPJ = OJQ BY SAS CONGRUENCE SO OP=OQ (C.P.C.T EQ 1 ) SIMARLY IN OQK AND OKR WE HAVE OQ =OR ( C.P.C.T) EQ2 NOW FROM EQ 1 AND 2 WE GET OP=OQ=OR
- 18. IF RADIUS ARE EQUAL โข SO THEN THE CIRCLE WILL PASS THROUGH THREE NON COLLINEAR POINTS โข HENCE PROVED
- 19. THE ANGLE SUBTENDED BY ARC OF A CIRCLE AT CENTRE IS DOUBLE THE ANGLE SUBTENDED BY IT AT ANY POINT ON THE REMAINING PART OF CIRCLE
- 20. SOL= GIVEN , AN MINOR PQ OF A CIRCLE AND A POINT R ON REMAINING PART OF CIRCLE i.e ARC QP TO PROVE POQ=2 PRQ CONS=JOIN RO PRODUCE PROOF = IN TRIANGLE ROP PO=RO (RADII OF CIRCLE ) SO POR IS ISOSCELES TRIANGLE SO RPO =PRO =x (IF SIDES ARE EQUAL ANGLES ARE EQUAL )
- 21. โข OQ= OR (RADII OF CIRCLE ) โข SO ROQ IS ISOSCELES TRIANGLE โข ORQ = OQR (IF SIDES ARE EQUAL ANGLES ARE EQUAL) โข POM IS EXTERIOR ANGLE OF RPO SO RPO + PRO = POM โข x+x = POM (exterior angle of triangle is equal to sum of two interior angles) 2x = POM( EQ1) In ROQ
- 22. Q โข NOW IN POQ โข ORQ +RQO = QOM (exterior angle of triangle is equal to sum of two interior angles) 2Y =QOM(EQ 2) โข FROM EQ 1 AND2 WE GET โข 2X+2Y =POM +QOM โข 2(X+Y)= POQ โข PRQ=1/2 POQ โข HENCE PROVED
- 23. THANK YOU