10.3 circles
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The document defines circles and provides methods for graphing, writing equations of, and finding tangent lines to circles. It defines a circle as all points equidistant from the center and gives the radius as the distance from the center to any point on the circle. Graphing circles involves writing the standard form equation and plotting points a radius distance from the origin. Writing the equation of a circle given a point involves using the distance formula to find the radius and plugging into the standard form. Finding tangent lines involves finding the slope of the radius and using the perpendicular slope and point-slope form.
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The document discusses key terms and theorems related to circles, including:
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Here are the key steps: instead of r and d
1. Read the site information carefully. Pay attention to the definitions and formulas.
2. Study the examples to understand how to apply the formulas.
3. Read the directions for the practice questions carefully.
4. Work through the practice questions, checking your work.
5. If you get a question wrong, review the relevant example(s) and try the question again.
6. Keep working through questions until you understand the concepts and can get them correct.
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4. Work through the practice questions, checking your work.
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3. Diagrams are included labeling the parts of a circle like the radius, diameter, chord, secant, and tangent. Definitions are provided for each term.
mathemaics 10 lesson about cicles. its part
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the circle and its parts.pptx
The document discusses the different parts of a circle, including radii, diameters, chords, secants, tangents, arcs, central angles, and inscribed angles. It provides examples and definitions for each part. The document emphasizes that understanding these parts is essential for solving problems involving measuring arcs of circles. It presents examples measuring arcs using properties like the central angle theorem. Finally, it provides practice problems for students to demonstrate their understanding of circle geometry concepts.
13534.ppt
This document defines and explains the key terms used to describe the different parts of a circle:
- The circumference is the distance around the outside of the circle. The diameter is the distance from one side to the other passing through the center.
- The radius is the line connecting the center of the circle to the circumference. A chord is a line touching the circumference at two points.
- A sector is the part of a circle between two radii. An arc is the part of the circumference at the edge of a sector. A segment is the part between a chord and an arc.
- A tangent is a straight line touching the circle at just one point. The document includes diagrams labeling these different parts
13534.ppt
This document defines and explains the key terms used to describe the different parts of a circle:
- Circumference is the distance around the outside of a circle. Diameter is the distance from one side to the other passing through the center. Radius is the line connecting the center to the circumference.
- A chord is a line touching the circumference at two points. An arc is part of the circumference at the edge of a sector. A sector is the part of a circle between two radii.
- A segment is the part of a circle between a chord and an arc. A tangent is a straight line touching the circle at one point.
The document includes diagrams labeling these terms and example problems applying
13534.ppt
This document defines and explains key terms related to parts of a circle:
- The circumference is the distance around the outside of a circle. The diameter is the distance from one side to the other passing through the center.
- The radius is the line connecting the center of the circle to the circumference. A chord is a line touching the circumference at two points.
- A sector is the part of a circle between two radii and an arc. An arc is the part of the circumference at the edge of a sector.
- A segment is the part of a circle between a chord and an arc. A tangent is a straight line touching a circle at one point.
13534.ppt
This document defines and explains the key terms used to describe the different parts of a circle:
- Circumference is the distance around the outside of a circle. Diameter is the distance from one side to the other passing through the center. Radius is the line connecting the center to the circumference.
- A chord is a line touching the circumference at two points. An arc is part of the circumference at the edge of a sector. A sector is the part of a circle between two radii.
- A segment is the part of a circle between a chord and an arc. A tangent is a straight line touching the circle at one point.
The document includes diagrams labeling these terms and example problems applying
13534.ppt
This document defines and explains key terms related to parts of a circle:
- The circumference is the distance around the outside of a circle. The diameter is the distance from one side to the other passing through the center.
- The radius is the line connecting the center of the circle to the circumference. A chord is a line touching the circumference at two points.
- A sector is the part of a circle between two radii and an arc. An arc is the part of the circumference at the edge of a sector.
- A segment is the part of a circle between a chord and an arc. A tangent is a straight line touching a circle at one point.
Secants and Tangents of Circles PowerPoint.ppt
The document discusses secants, tangents, and tangent circles. It defines a secant as a line intersecting a circle at two points, and a tangent as a line intersecting at one point. Tangent lines are perpendicular to the radius at the point of contact. Tangent and secant segments are parts of these lines. Two tangent circles intersect at one point, which lies on the line joining their centers. Common tangents are lines tangent to two circles, either internally or externally. The document provides examples and problems involving tangent lines and circles.
G10 Math Q2- Week 8- Equation of a Circle.pptx
This document discusses the standard and general forms of equations for circles. It provides examples of writing the standard and general form equations given the center and radius of various circles. Key points covered include:
- The standard form of a circle equation is (x-h)2 + (y-k)2 = r2, where (h,k) is the center and r is the radius.
- The general form is Ax2 + By2 + Cx + Dy + E = 0, where the coefficients can be used to find the center and radius.
- Examples are worked through of writing circle equations in both standard and general form given attributes of the circle like the center and radius.
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.
2. PARABOLA (PRECAL).pptx
This document discusses parabolas including:
- Defining a parabola as points equidistant from a fixed point (focus) and fixed line (directrix)
- Key parts of a parabola including the vertex, focus, directrix, axis of symmetry, and latus rectum
- The standard forms of equations for parabolas with the vertex at the origin, depending on the orientation
- Characteristics that can be determined from a parabola equation in standard form, such as the vertex, focus, directrix, latus rectum and its endpoints
- Examples of finding the equation of a parabola given characteristics like the focus or directrix
Pre-Cal 40S Slides December 11, 2007
This document discusses parabolas, circles, and ellipses. It provides instructions on how to sketch these shapes from their equations, identifying features like the vertex, focus, axis of symmetry, and directrix for parabolas. For circles, it asks the reader to determine the standard form equation, radius, and center coordinates. The document explores transforming between the standard and general forms of parabolas and circles.
Circlesppt
This document provides definitions and characteristics of circles. It defines a circle as all points equidistant from a center point, and defines related terms like radius, diameter, chord, tangent, secant, arc, and circumference. It explains properties of circles including pi, inscribed angles, intercepted arcs, and provides examples and non-examples of parts of a circle.
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Circle - Basic Introduction to circle for class 10th maths.
Circle - Basic Introduction to circle for class 10th maths.
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7.3 rational exponents
Nth roots can be expressed using rational exponents, where the denominator of the exponent is the index of the radical. To evaluate expressions with rational exponents, one should rewrite them in radical form, evaluate the radical, and then raise the expression to the power indicated by the original exponent. Examples are provided of evaluating various expressions with rational exponents.
6.6 quadratic formula
The document introduces the quadratic formula as a method for solving any quadratic equation in the standard form of ax^2 + bx + c = 0. It provides the quadratic formula, and shows an example of using it to solve for x in a quadratic equation. Users are then prompted to try solving for x in a quadratic equation using the quadratic formula.
6 7 new look at conics
This document discusses classifying second-degree equations as different types of conic sections. It provides the general form of a second-degree equation, explains how to use the discriminant to determine if the graph is a circle, ellipse, parabola, or hyperbola, and gives examples of identifying the conic section from different equations. Key aspects covered include degenerate conics, the general form containing terms for x2, xy, y2, x, y, and the constant, and using the discriminant calculated from the equation coefficients to classify the conic section.
6 6 systems of second degree equations
This document discusses methods for solving systems of second-degree equations, including algebraic substitution or elimination methods and graphical methods. It provides 3 examples of solving systems of equations algebraically and graphically, including solving systems of quadratic equations and systems involving circles.
4.4 multi step trig problems
This document provides instructions for solving for the length of a shared side (x) of two triangles. It instructs the user to write two equations, one for each triangle, relating the lengths of the sides including x. It also describes an approach for solving when the triangles form one large non-right triangle, which is to draw the altitude splitting it into two triangles and using the fact that triangle angles sum to 180 degrees. The user is then told to solve the two equations to find the value of x.
6 5 parabolas
This document provides an overview of parabolas including their key properties and examples of how to graph and write equations of parabolas. It defines that every point on a parabola is equidistant from the focus and directrix, with the directrix perpendicular to the line of symmetry and the vertex halfway between them. Examples are given of finding the focus, directrix, and graphing parabolas given these features as well as writing the equation of a parabola given its directrix.
4.3 finding missing angles
Inverse trig ratios, including sin-1, cos-1, and tan-1, are used to solve for missing angle measures. When given a decimal value for sin, cos, or tan, the corresponding inverse trig function is used to find the angle measure to the nearest degree. When given a triangle, the appropriate trig ratio is set up and the inverse function is then used to find the measure of the indicated angle to the nearest degree.
4.2 solving for missing sides
This document provides instructions for solving for a missing side of a triangle using trigonometric functions. It tells the reader to label the sides of the triangle, set up an equation using the information given, and solve for the variable to find the missing side, making sure a calculator is in degree mode.
4.1 trig ratios
This document defines and explains the three main trigonometric ratios - sine, cosine, and tangent - which are used to solve for missing angles and sides of right triangles. It identifies the hypotenuse as the longest side, and the adjacent and opposite sides in relation to a given angle. The document provides the ratio equations and examples of calculating trig ratios for various angles in a right triangle.
R.4 solving literal equations
This document defines a literal equation as an equation with two or more different variables. It provides examples of literal equations such as A = bh, where b is solved for, and P = 2L + 2W, where W is solved for. The document demonstrates how to solve various literal equations for the indicated variable, such as solving for P in I = Prt, solving for x in A = x + y^2, and solving for x and y in two additional equations.
R.3 solving 1 var inequalities
This document provides instructions for solving inequalities and graphing their solutions. It notes that when multiplying or dividing an inequality by a negative number, the inequality symbol is reversed. Open circles are used for < and >, while closed circles are used for ≤ and ≥. Examples are provided of solving various inequalities algebraically and graphing the resulting solution sets on a number line.
R.2 solving multi step equations
Multi-step equations involve more than two steps to solve. To solve them, simplify the side with variables by distributing and combining like terms, then solve as a single-step or two-step equation. The key steps are to distribute any parentheses, collect like variable terms on one side and constants on the other, and use opposite operations to solve for the variable.
R.1 simplifying expressions
Algebraic expressions contain numbers, variables, and operators and can be simplified using the distributive property and by combining like terms. The document provides examples of algebraic expressions that can be simplified by first distributing all terms inside parentheses and then combining like terms, such as 3(4n + 2) + n, which is simplified to 12n + 6 + n = 13n + 6.
10 4 solving trig equations
This document discusses two methods for solving trigonometric equations: graphing and algebraically. Graphing allows for visually seeing solutions but may result in decimal answers, while algebraic solving provides exact solutions but can become messy. It provides tips for algebraic solving based on whether the equation contains functions of 2x and x, or only 2x. Examples are given to demonstrate each method.
7 2 adding and subtracting polynomials
The document discusses several methods for adding and subtracting polynomials: using algebra tiles, the horizontal method, and the vertical method. It provides examples of adding and subtracting polynomials with one and two variables. Terms with the same variables are combined by adding the coefficients. For subtraction, the signs of the terms in the subtracted polynomial are changed before adding.
7 3 multiplying polynomials
This document discusses four methods for multiplying binomial expressions: algebra tiles, the distributive property, a table or "box" method, and FOIL. It explains that every term in the first binomial must be multiplied by every term in the second binomial. Examples are provided of multiplying binomials such as (x + 2)(x + 5), along with practice problems like (x + 3)(x - 4) and (2p - 4)(3p + 2).
7.2 simplifying radicals
The document discusses simplifying radical expressions. It defines key terms like the radical sign and radicand. It provides two methods for simplifying radicals: using the product property to rewrite with the largest perfect square factor or making a factor tree to pull out factors. Examples are provided to demonstrate simplifying radicals of various forms, including those with variables and higher root expressions. The document also contains practice problems for simplifying radical expressions.
10 3 double and half-angle formulas
1) This document discusses double-angle and half-angle formulas for trigonometric functions like sine, cosine, and tangent.
2) It derives formulas that relate trig functions of double and half angles to trig functions of the original angle, such as sin(2x) = 2sin(x)cos(x) and sin(x/2) = ±√(1-cos(x))/2.
3) Examples are provided to demonstrate applying these formulas to simplify trigonometric expressions and derive new identities.
6.7 other methods for solving
This document discusses two additional methods for solving quadratic equations: the quadratic formula and graphing. The quadratic formula can be used to solve any quadratic equation in standard form (ax^2 + bx + c = 0) by inputting the a, b, and c coefficients. Graphing involves changing the equation to y= form, plotting it on a calculator to find the x-intercepts, which are the solutions. Both methods are demonstrated with examples.
10 2 sum and diff formulas for tangent
1) Formulas are derived for tangent of the sum and difference of two angles using previous trigonometric identities: tan(α + β) = (tanα + tanβ) / (1 - tanαtanβ) and tan(α - β) = (tanα - tanβ) / (1 + tanαtanβ).
2) These formulas are valid when the tangents of the individual angles and their sum/difference are defined, which excludes cases where the cosine of any angle is 0.
3) Examples are provided to demonstrate applying these formulas to find the exact value of tangents of sums/differences of known angles.
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10.3 circles
- 2. Definition of a Circle All points on a circle are equidistant from the center of the circle. The distance between the center and any point on the circle is the radius.
- 3. Equation of a Circle Can be derived from the distance formula. Only for circles with center at (0, 0)!
- 4. Graphing Circles 1. Write in standard form. 2. Find the radius. 3. Plot four intercepts r units from the origin. 4. Connect with a circle. Example: Graph
- 6. Writing Equations 1. Use the distance formula to find r. 2. Plug it into standard form equation. Example: The point (1, 4) is on a circle centered at (0,0). Write the equation of the circle.
- 7. Your Turn! The point (5, 1) is on a circle centered at (0,0). Write the equation of the circle.
- 8. Tangent Lines A line is tangent to a circle if it touches the circle at one point and is perpendicular to the radius.
- 9. To find the equation: 1. Find the slope of the radius through the given point. 2. Find slope perpendicular to that (opp. reciprocal) 3. Use point-slope form to write equation.
- 10. Example Write an equation of the line tangent to the circle at (2, 3).
- 11. Your Turn! Write an equation of the line tangent to the circle at (-2, 4).