The document is a student's report on circles and geometry that includes definitions of key circle terms like radius, diameter, chord, and arc. It also proves several circle theorems, such as equal chords subtending equal angles at the center and the perpendicular from the center bisecting a chord. Additionally, it discusses drawing a circle through three points and the relationship between arcs and their corresponding chords. The report aims to explain fundamental circle concepts and their properties.
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.
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.
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 👕 and go for the world 🌎 to be a t y to the to do the world i the world 🌎 and pink y to y y u can you must i and white 🤍 to be a I will be in and pink rod is Shreya aur Priyanshi and resume tow bar 🍺 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 🌍 to go to go with the world of interest in the morning 🌄🌄🌄🌄🌄🌄🌄🌅🌅🌅 to be a good mood to this email ✉️✉️📨 to go to go to go with you and your help in the morning 🌄 to be in and pink y to y y u yum 😋😋😋😋😋😋😋😋😋 to go to go to go to this email 📨📨📨 to go to be a good 😊😊😊 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 📸📸📸📸📸 to go to reel featuring David and resume for two 🕑🕑🕑 to go I have to check the to you y u sir to be in touch and white 🐻❄️🐻❄️ to be of any kind and white 🐻❄️🐻❄️ to go with you and your help and white 🐻❄️🐻❄️🐻❄️🐻❄️🐻❄️🐻❄️ 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 📸📸📸 to go with you and pink y u sir to send the measure to be of interest in our country and white 🐻❄️🐻❄️🐻❄️🐻❄️ to go to you y y to y u yum 😋😋 to go to m go to sleep 😴😴😴😴😴 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 🕑🕑🕑🕑 to be a little t to a t shirt 🎽🎽🎽🎽 to go to go to go to go to school 🏫🏫🏫 to be a little t y t to go to learn y u can be of SRP to be in the world 🌍🌍 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 😊😊😊😊😊😊😊 to be of SRP to be in and out go talking with a t shirt 🎽🎽🎽 to go to go to go with a good 😊😊😊 to go to go to go to go with you must i to go with the measure of interest in the morning 🌅🌅 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 😊😊😊😊😊 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 💤😴😴😴 and resume for your help to learn to learn by you and your family a very happy birthday dear fr
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.
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.
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.
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.
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.
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.
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.
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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.
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.
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.
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.
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.
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.
- 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.
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.
Grade 9 (Alternate) Mathematics III - Learning Modules for EASE Program of DepEdR Borres
Grade 9 (Alternate) Mathematics III - Learning Modules for EASE Program of DepEd
A compilation of Math III learning modules for EASE which can be alternate for Grade 9 Mathematics.
Free!
This document contains examples and exercises about circles from a math textbook. It defines key terms like radius, diameter, chord, arc, and sector. It includes questions about properties of circles like equal chords subtending equal angles, relationships between intersecting circles, and finding lengths and distances related to circles. For example, one question asks the reader to prove that if two equal chords intersect within a circle, the segments of one chord are equal to the corresponding segments of the other chord.
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.
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=2×radius.
- Examples are given to demonstrate determining if points lie inside, outside, or on the circle.
1. This module discusses characteristics of circles such as lines, segments, arcs, and angles. It defines circles and their components like radii, chords, diameters, secants, and tangents.
2. The module covers relationships between these components, such as a radius bisecting a chord if it is perpendicular to it. It also defines types of arcs and angles, such as central angles that are equal to their intercepted arcs.
3. The summary provides examples of applying theorems about congruent arcs, chords, and angles to determine if components are congruent in circles or congruent circles.
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.
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.
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
1. The document defines various terms related to circles such as chord, diameter, secant, tangent, and tangent segment. It also states some properties of these terms like a secant containing a chord and a tangent segment joining any point on a tangent to the point of contact.
2. Several theorems related to circles are presented, such as a tangent being perpendicular to the radius at the point of contact, and the lengths of two tangent segments drawn from an external point to a circle being equal.
3. Proofs of some circle theorems are shown, for example proving that a line perpendicular to the radius at its outer end is a tangent to the circle.
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.
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.
The document discusses circles, defining them as sets of points equidistant from a center point. It describes key circle terms like diameter, radius, chord, and circumference. Formulas are provided relating circumference to diameter using pi, diameter to radius, and area to radius. Examples demonstrate calculating circumference from diameter, diameter from circumference, and area from radius using the formulas. The document aims to define and explain key geometric concepts relating to circles through definitions, explanations, and example calculations.
The document discusses circles, defining them as sets of points equidistant from a center point. It describes key circle terms like diameter, radius, chord, and circumference. Formulas are provided relating circumference to diameter using pi, diameter to radius, and area to radius. Examples demonstrate calculating circumference from diameter, diameter from circumference, and area from radius using the formulas. The document aims to define and explain key geometric concepts relating to circles through definitions, explanations, and example calculations.
The document discusses properties and theorems related to circles. It defines key circle concepts like radius, diameter, chord, and defines a circle as all points equidistant from a central point. It then lists several theorems about circles, such as equal chords subtend equal angles, perpendiculars from the center bisect chords, there is one unique circle through three non-collinear points, and angles in the same arc are equal. It also discusses properties of cyclic quadrilaterals where the four vertices all lie on the same circle.
This document discusses circle geometry and contains the following key points:
- It defines important parts of a circle like radii, chords, diameters, and arcs.
- It establishes theorems relating radii and diameters, congruent arcs and angles, congruent chords and arcs, and the relationships between chords, arcs, and diameters.
- It describes how to calculate arc measures, circumference, area of circles and sectors, and properties of inscribed angles and polygons.
This presentation discusses geometric shapes and spaces, specifically circles. It covers basic circle terms like radius, diameter, arc, chord, and sector. The document then explains several circle theorems regarding tangents, arcs and central angles, inscribed angles, and relationships between angles and intercepted arcs. Examples are provided to demonstrate how to use the theorems to find measures of angles. In the examples, statements and reasons are written to show the step-by-step work and logic. The presentation concludes by relating the measures of central angles to arc lengths and sector areas using formulas.
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.
- 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.
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.
Grade 9 (Alternate) Mathematics III - Learning Modules for EASE Program of DepEdR Borres
Grade 9 (Alternate) Mathematics III - Learning Modules for EASE Program of DepEd
A compilation of Math III learning modules for EASE which can be alternate for Grade 9 Mathematics.
Free!
This document contains examples and exercises about circles from a math textbook. It defines key terms like radius, diameter, chord, arc, and sector. It includes questions about properties of circles like equal chords subtending equal angles, relationships between intersecting circles, and finding lengths and distances related to circles. For example, one question asks the reader to prove that if two equal chords intersect within a circle, the segments of one chord are equal to the corresponding segments of the other chord.
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.
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=2×radius.
- Examples are given to demonstrate determining if points lie inside, outside, or on the circle.
1. This module discusses characteristics of circles such as lines, segments, arcs, and angles. It defines circles and their components like radii, chords, diameters, secants, and tangents.
2. The module covers relationships between these components, such as a radius bisecting a chord if it is perpendicular to it. It also defines types of arcs and angles, such as central angles that are equal to their intercepted arcs.
3. The summary provides examples of applying theorems about congruent arcs, chords, and angles to determine if components are congruent in circles or congruent circles.
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.
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.
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
1. The document defines various terms related to circles such as chord, diameter, secant, tangent, and tangent segment. It also states some properties of these terms like a secant containing a chord and a tangent segment joining any point on a tangent to the point of contact.
2. Several theorems related to circles are presented, such as a tangent being perpendicular to the radius at the point of contact, and the lengths of two tangent segments drawn from an external point to a circle being equal.
3. Proofs of some circle theorems are shown, for example proving that a line perpendicular to the radius at its outer end is a tangent to the circle.
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.
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.
The document discusses circles, defining them as sets of points equidistant from a center point. It describes key circle terms like diameter, radius, chord, and circumference. Formulas are provided relating circumference to diameter using pi, diameter to radius, and area to radius. Examples demonstrate calculating circumference from diameter, diameter from circumference, and area from radius using the formulas. The document aims to define and explain key geometric concepts relating to circles through definitions, explanations, and example calculations.
The document discusses circles, defining them as sets of points equidistant from a center point. It describes key circle terms like diameter, radius, chord, and circumference. Formulas are provided relating circumference to diameter using pi, diameter to radius, and area to radius. Examples demonstrate calculating circumference from diameter, diameter from circumference, and area from radius using the formulas. The document aims to define and explain key geometric concepts relating to circles through definitions, explanations, and example calculations.
The document discusses properties and theorems related to circles. It defines key circle concepts like radius, diameter, chord, and defines a circle as all points equidistant from a central point. It then lists several theorems about circles, such as equal chords subtend equal angles, perpendiculars from the center bisect chords, there is one unique circle through three non-collinear points, and angles in the same arc are equal. It also discusses properties of cyclic quadrilaterals where the four vertices all lie on the same circle.
This document discusses circle geometry and contains the following key points:
- It defines important parts of a circle like radii, chords, diameters, and arcs.
- It establishes theorems relating radii and diameters, congruent arcs and angles, congruent chords and arcs, and the relationships between chords, arcs, and diameters.
- It describes how to calculate arc measures, circumference, area of circles and sectors, and properties of inscribed angles and polygons.
This presentation discusses geometric shapes and spaces, specifically circles. It covers basic circle terms like radius, diameter, arc, chord, and sector. The document then explains several circle theorems regarding tangents, arcs and central angles, inscribed angles, and relationships between angles and intercepted arcs. Examples are provided to demonstrate how to use the theorems to find measures of angles. In the examples, statements and reasons are written to show the step-by-step work and logic. The presentation concludes by relating the measures of central angles to arc lengths and sector areas using formulas.
Similar to Geometry - Mathematics - 9th Grade by Slidesgo.pptx (20)
How to Fix the Import Error in the Odoo 17Celine George
An import error occurs when a program fails to import a module or library, disrupting its execution. In languages like Python, this issue arises when the specified module cannot be found or accessed, hindering the program's functionality. Resolving import errors is crucial for maintaining smooth software operation and uninterrupted development processes.
বাংলাদেশের অর্থনৈতিক সমীক্ষা ২০২৪ [Bangladesh Economic Review 2024 Bangla.pdf] কম্পিউটার , ট্যাব ও স্মার্ট ফোন ভার্সন সহ সম্পূর্ণ বাংলা ই-বুক বা pdf বই " সুচিপত্র ...বুকমার্ক মেনু 🔖 ও হাইপার লিংক মেনু 📝👆 যুক্ত ..
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LAND USE LAND COVER AND NDVI OF MIRZAPUR DISTRICT, UPRAHUL
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.
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.
How to Manage Your Lost Opportunities in Odoo 17 CRMCeline George
Odoo 17 CRM allows us to track why we lose sales opportunities with "Lost Reasons." This helps analyze our sales process and identify areas for improvement. Here's how to configure lost reasons in Odoo 17 CRM
How to Add Chatter in the odoo 17 ERP ModuleCeline George
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.
A workshop hosted by the South African Journal of Science aimed at postgraduate students and early career researchers with little or no experience in writing and publishing journal articles.
2. Table Of Contents
1. Circles And Its Related Terms.
2. Angle Subtended By A Chord At A Point.
3. Perpendicular From The Centre To A Chord.
4. Circle Through Three Points
5. Equal Chords And Their Distances From The Centre.
6. Angle Subtended By An Arc Of A Circle.
7. Cyclic Quadrilaterals.
8. What We Conclude?
For
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3. Introduction:
We come across many objects in our daily life which are round in
shape, such as 1 rupee coin, wheels of vehicles, bangles etc. How can
we define a circle? A circle is a closed figure in a plane and it is the
collection of all the points in the plane which are at a constant
distance from a fixed point in the plane. The fixed point is the center
of the circle and the constant distance is the radius of the circle.
4. Circles and its related terms
Circumference of a circle is the length of the complete circular curve
constituting the circle.
Chord of a circle is a line segment joining any two points on the
circle.
Arc of a circle is a part of the circle. Any two points A and B of a circle
divide the circle into two parts.
The smaller part is called the minor arc and the larger part is called
the major arc of the circle.
If the two parts are equal, AB is a diameter of the circle and each part
is called a semi circle.
5. Angle Subtended By A Chord At A Point:
We take a line segment PQ and a point R not on the
line containing PQ. Join PR and QR. Then ∠PRQ is
called the angle subtended by the line segment PQ at
the point R.ZPOQ is the angle subtended by the chord
PQ at the center O, ZPRQ and ZPSQ are respectively
the angles subtended by PQ at points R and S on the
major and minor arcs PQ
6. THEOREM 1
To Prove: Equal chords of a circle subtend equal angles at the center.
Proof: We are given two equal chords AB and CD of a circle with center O. We want to
prove that: Angle AOB = Angle COD.
In triangles AOB and COD,
OA = OC (Radii of a circle)
OB = OD (Radii of a circle)
AB = CD (Given)
Therefore, Triangle AOB = Triangle COD (SSS rule)
This gives ∠AOB = COD (CPCT)
;
7. If the angles subtended by the chords of a circle at
the center are equal, then the chords are equal.
Theorem 2 is converse of the Theorem 1.
THEORUM 2
8. Perpendicular From The Centre To A Chord:
p
o
Let there be a circle with O as center and let AB one of its chord.
Draw a perpendicular through O cutting AB at M .
Then , ZOMA LOMB = 90° or OM is perpendicular to AB.
We conclude that the perpendicular from the center of a
circle to a chord bisects the chord.
9. THEOREM 3
TO PROVE: The line drawn through the center of a circle to bisect a chord
is perpendicular to the chord.
PROOF: Let AB be a chord of a circle with center O and O is joined to the
mid-point M of AB.
We have to prove that OM perpendicular to AB.
Join OA and OB. In triangles OAM and OBM,
OA = OB (Given)
AM = BM (Given)
OM = OM (Common)
Therefore, ΔΟΑΜ = ΔΟΒΜ (SSS)
This gives angle OMA = angle LOMB = 90° (CPCT)
10. CIRCLE THROUGH THREE POINTS:
Infinite number of circle can be drawn through one or
two points (Fig 1 And 2).
In case there are three collinear points, then circle can
be drawn but the third point will lie outside the circle
(Fig 3).
Let us take three points A, B and C, which are not on the
same line (Fig 4).
Draw perpendicular bisectors of AB and BC, PQ and RS
respectively Let these perpendicular bisectors intersect
at one point O (Fig 5).
12. Equal Chords And Their Distances From The Centre:
Let AB be a line and P be a point. Since there are infinite
numbers of points on a line, if we join these points to P, we
will get infinitely many line segments PL1, PL2, PM, PL3, PL4,
etc.
Out of these line segments, the perpendicular from P to AB,
namely PM will be the least.
We conclude that the length of the perpendicular from a point
to a line is the distance of the line from the point P.
13. Theorum 4
TO PROVE: Equal chords of a circle (or of congruent circles) are equidistant from the center (or
centers).
PROOF: Take a circle of any radius. Draw two equal chords AB and CD of it and also the
perpendiculars OM and ON on them from the center O.
Divide the figure into two so that D falls on B and C falls on A . We observe that O lies on the
crease and N falls on M.
Therefore, OM = ON.
14. Theorem 5
Chords equidistant from the center
of a circle are equal in length.
Theorem 5 is converse of the
Theorem 4
15. Angle Subtended By An Arc Of A Circle:
The end points of a chord other than diameter of a circle cuts it into two arcs-one
major and other minor.
If we take two equal chords, They are more than just equal in length. They are
congruent in the sense that if one arc is put on the other, without bending or
twisting, one superimposes the other completely.
We can verify this fact by cutting the arc, corresponding to the chord CD from the
circle along CD and put it on the corresponding arc made by equal chord AB. We
will find that the arc CD superimpose the arc AB completely This shows that equal
chords make congruent arcs and conversely congruent arcs make equal chords of
a circle.
We conclude that if two chords of a circle are equal, then their corresponding arcs
are congruent and conversely, if two arcs are congruent, then their corresponding
chords are equal.
16. Theorem 6
TO PROVE: The angle subtended by an arc at the center is double the angle
subtended by it at any point on the remaining part of the circle.
PROOF: Given an arc PQ of a circle subtending angles POQ at the center O and
PAQ at a point A on the remaining part of the chord
Consider the three different cases arc PQ is minor (Fig 1), arc PQ is a
semicircle(Fig 2) and in arc PQ is major fig.3
Let us begin by joining AD and extending it to a point B all the cases,
angle BOQ= angle OAQ + angle AQO because an exterior angle of a triangle is
equal to the sum of the two interior opposite angles.
In ΔOAQ,
OQ=OA [Radii of a circle]
Therefore, angle OAQ = angleQOA
This gives, angle BOQ=2(angleOAQ)
Similarly, angle BOP =2 angle OAP (2)
From (1) and (2), angle BOP + angle BOQ=2( angle OAP+ angle OAQ)
This is the same as angle POQ =2 angle PAQ (3)
For the case (iii), where PQ is the major arc, (3) is replaced by reflex angle POQ
= 2 < PAQ.
17. Jupiter's rotation period
9h 55m 23s
Distance between Earth and the Moon
386,000 km
The Sun’s mass compared to Earth’s
333,000
18. Student’s geometry performance
Venus is beautiful and the second-
brightest natural object in the
night sky after the Moon
70%
Engaging in activities
Mercury is also the closest planet
to the Sun and the smallest in the
entire Solar System
50%
Score improvement
19. Our schools
School 1
Jupiter was named after
the god of the skies
School 3
Despite being red, Mars is
actually a cold place
School 2
Venus is the second
planet from the Sun
20. History of geometry
Euclidean geometry Despite being red, Mars is
actually a cold place
300 BCE
Non-euclidean Jupiter was named after the
Roman god of the skies
19th CE
Pythagorean theorem Earth is the third planet
from the Sun and has life
6th CE
Archimedes Venus has a beautiful
name, but it’s hot
3rd CE
21. Important geometry formulas
Despite being red, Mars is actually a
cold place. It’s full of iron oxide dust,
which gives the planet its reddish cast
c2=a2+b2
Pythagorean theorem
Earth is the third planet from the Sun
and the only one that harbors life in
the Solar System
A=½ x base x height
Area of a triangle
22. Properties of geometry shapes
Shape Definition Properties
Triangle A 3-sided polygon
⃞ We all live on Earth
⃞ Mercury is very small
Quadrilateral A 4-sided polygon
⃞ Venus is a hot planet
⃞ Mars is a cold planet
Circle
A round shape
with constant radius
⃞ Jupiter is a gas giant
⃞ Saturn has rings
Polygon
A closed, flat shape with
straight sides
⃞ Neptune is an ice giant
⃞ Earth is a blue planet
23. Follow the link in the graph to modify its data and then paste the new one here. For more info, click here
Ways to improve geometry comprehension
Practice
Venus is the second
planet from the Sun
61%
Visualize
Earth is also known as
the Blue Planet
57%
Examples
Despite being red, Mars
is actually a cold place
43%
Ask questions
Jupiter was named after
the god of the skies
22%
24. Our teachers
You can speak a bit about
this person here
Jenna Doe
You can speak a bit about
this person here
Susan Bones
You can speak a bit about
this person here
Timmy Jimmy
25. Constructing shapes
Use the lines provide to make a pentagon. You can rotate them if necessary
Lines My pentagon
27. Area and perimeter quiz
Calculate the area and perimeter of the given shapes below. Type your answer in the text box
Area Write your answer here
Perimeter Write your answer here
Side A:
Side B:
Side C:
9 cm
7 cm
12 cm
Area Write your answer here
Perimeter Write your answer here
Length:
Width:
10 cm
6 cm
28. Problem solving methods
Saturn is also a gas giant
and the biggest planet in the
Solar System
Proportions
Mars is full of iron oxide
dust, giving the planet its
reddish cast
Ratios
Mercury is the closest planet
to the Sun and the smallest
in the System
Theorems
Venus is the second-
brightest natural object in
the night sky
Diagrams
Jupiter is a gas giant and the
biggest planet in the entire
Solar System
Similarity
29. Conclusions
This planet's name has nothing to do with the
liquid metal, since Mercury was named after
the Roman messenger god
⃞ Venus has a toxic atmosphere
⃞ Earth is the third planet from the Sun
⃞ Mars was named after a god
30. Credits: This presentation template was created by
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infographics & images by Freepik.
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32. Alternative resources
Vectors
⃞ Geometry and maths graphs
⃞ Maths realistic chalkboard background
Here’s an assortment of alternative resources whose style fits the one of this template
33. Resources
Vectors
⃞ Geometry and maths graphs
⃞ Maths realistic chalkboard background
Photos
⃞ Classmates sitting and learning at desk
⃞ Two teenage girls studying together at
home on laptop
⃞ Smiling portrait of a young
businesswoman standing behind the desk
against white wall
⃞ Portrait of a happy businessman
standing in office
⃞ Smiling african american young lady
near window
Icons
⃞ Icon pack: geometry | Filled
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