Circles
Download
1 / 52

Circles - PowerPoint PPT Presentation


  • 111 Views
  • Uploaded on

Circles. Chapter 10. 10.1 Tangents to Circles. Circle : the set of all points in a plane that are equidistant from a given point. Center : the given point. Radius : a segment whose endpoints are the center of the circle and a point on the circle. Vocabulary.

loader
I am the owner, or an agent authorized to act on behalf of the owner, of the copyrighted work described.
capcha
Download Presentation

PowerPoint Slideshow about ' Circles' - jett


An Image/Link below is provided (as is) to download presentation

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.


- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -
Presentation Transcript
Circles

Circles

Chapter 10


10 1 tangents to circles
10.1 Tangents to Circles

  • Circle: the set of all points in a plane that are equidistant from a given point.

  • Center: the given point.

  • Radius: a segment whose endpoints are the center of the circle and a point on the circle.


Vocabulary
Vocabulary

  • Chord: a segment whose endpoints are points on the circle.

  • Diameter: a chord that passes through the center of the circle.

  • Secant: a line that intersects a circle in two points.

  • Tangent: a line in the plane of a circle that intersects the circle in exactly one point.


More vocabulary
More Vocabulary

  • Congruent Circles: two circles that have the same radius.

  • Concentric Circles: two circles that share the same center.

  • Tangent Circles: two circles that intersect in one point.


Tangent theorems
Tangent Theorems

  • If a line is tangent to a circle, then it is perpendicular to the radius drawn to the point of tangency.

P

Q


Tangent theorems1
Tangent Theorems

  • In a plane, if a line is perpendicular to a radius of a circle at its endpoint on the circle, then the line is tangent to the circle.

P

Q


Tangent theorems2
Tangent Theorems

  • If two segments from the same exterior point are tangent to a circle then they are congruent.

Q

P


Examples
Examples

  • Find an example for each term:

    Center

    Chord

    Diameter

    Radius

    Point of Tangency

    Common external tangent

    Common internal tangent

    Secant


Examples1
Examples

  • The diameter is given. Find the radius.

    • d=15cm

    • d=6.7in

    • d=3ft

    • d=8cm


Examples2
Examples

  • The radius is given. Find the diameter.

    • r = 26in

    • r = 62ft

    • r = 8.7in

    • r = 4.4cm


Examples3
Examples

  • Tell whether AB is tangent to C.

A

14

5

B

15

C


Examples4
Examples

  • Tell whether AB is tangent to C.

A

12

C

16

8

B


Examples5
Examples

  • AB and AD are tangent to C. Find x.

D

2x + 7

A

5x - 8

B


10 2 arcs and chords
10.2 Arcs and Chords

  • An angle whose vertex is the center of a circle is a central angle.

  • If the measure of a central angle is less than 180 , then A, B and the points in the interior of APB form a

    minor arc.

  • Likewise, if it is greater

    than 180, if forms a major

    arc.


Arcs and chords
Arcs and Chords

  • If the endpoints of an arc are the endpoints of a diameter, then the arc is a semicircle.

  • The measure of an arc is the same as the measure of its central angle.


Examples6
Examples

  • Find the measure of each arc.

    • MN

    • MPN

    • PMN


Arc addition postulate
Arc Addition Postulate

  • The measure of an arc formed by two adjacent arcs is the sum of the measures of the two arcs.

    • mABC = mAB + mBC


Examples7
Examples

  • Find the measure of each arc.

    • GE

    • GEF

    • GF


Theorems
Theorems

  • In the same circle, or in congruent circles, congruent chords have congruent arcs and congruent arcs have congruent chords.

  • In the same circle, or in congruent circles, two chords are congruent iff they are equidistant from the center.


Examples8
Examples

  • Determine whether the arc is minor, major or a semicircle.

    • PQ

    • SU

    • QT

    • TUP

    • PUQ


Examples9
Examples

  • KN and JL are diameters. Find the indicated measures.

    • mKL

    • mMN

    • mMKN

    • mJML


Examples10
Examples

  • Find the value of x. Then find the measure of the red arc.


Homework
Homework

  • Pg. 600 # 26-28, 37, 39, 47, 48

  • Pg. 607 # 12-30 even, 32-34, 37-38


10 3 inscribed angles
10.3 Inscribed Angles

  • An inscribed angle is an angle whose vertex is on a circle and whose sides contain chords of the circle.

  • The arc that lies in the interior of an inscribed angle and has endpoints on the angle is called the intercepted arc.


Measure of an inscribed angle
Measure of an Inscribed Angle

  • If an angle is inscribed in a circle, then its measure is half the measure of its intercepted arc.

    • mADB = ½ mAB



Theorem
Theorem

  • If two inscribed angles if a circle intercept the same arc, then the angles are congruent.


Polygons and circles
Polygons and circles

  • If all of the vertices of a polygon lie on a circle, the polygon is inscribed in the circle, and the circle is circumscribed about the polygon.


Theorems1
Theorems

  • If a right triangle is inscribed in a circle, then the hypontenuse is a diameter of the circle.

    • B is a right angle iff AC is a diameter of the circle.


Theorems2
Theorems

  • A quadrilateral can be inscribed in a circle iff its opposite angles are supplementary.

D, E, F, and G lie on some circle, C, if and only if

mD + mF = 180° and mE + mG = 180°


Examples11
Examples

  • Find the value of each variable.


More examples
More Examples

  • Pg 616 # 2-8


10 4 other angle relationships in circles
10.4 Other Angle Relationships in Circles

  • If a tangent and a chord intersect at a point on a circle, then the measure of each angle formed is one half the measure of its intercepted arc.

    • m 1 = ½ mAB

    • m 2 = ½ mBCA


Examples12
Examples

  • Line m is tangent to the circle. Find the measure of the red angle or arc.


Examples13
Examples

  • BC is tangent to the circle. Find m CBD.


Theorems3
Theorems

  • If two chords intersect in the interior of a circle, then the measure of each angle is one half the sum of the measures of the arcs intercepted by the angle and its vertical angle.

    • x = ½ (mPS + mRQ)

    • x = ½ (106 + 174 )

    • x = 140


Theorems4
Theorems

B

A

  • m 1 = ½ (mBC – mAC)

  • m 2 = ½ (mPQR – mPR)

  • m 3 = ½ (mXY – mWZ)

1

C

P

2

Q

R

X

W

3

Z

Y


More examples1
More Examples

  • Pg. 624 #2-7


10 5 segment lengths in circles
10.5 Segment Lengths in Circles

  • When 2 chords intersect inside of a circle, each chord is divided into 2 segments, called segments of a chord.

  • When this happens, the product of the lengths of the segments of one chord is equal to the product of the lengths of the segments of the other chord.


Examples14
Examples

  • Find x.


Vocabulary1
Vocabulary

  • PS is a tangent segment because it is tangent to the circle at an endpoint.

  • PR is a secant segment and PQ is the external segment of PR.


Theorems5
Theorems

  • EA EB = EC ED

B

A

E

C

D


Theorems6
Theorems

  • (EA)2 = EC ED

  • EA is a tangent segment, ED is a secant segment.

A

E

C

D


Examples15
Examples

  • Find x.


Examples16
Examples

  • Find x.


Examples17

x ___ = 10 ___

X2 = 4 ____

Examples


10 6 equations of circles
10.6 Equations of Circles

  • You can write the equation of a circle in a coordinate plane if you know its radius and the coordinates of its center.

  • Suppose the radius is r and its center is at ( h, k)

    • (x – h)2 + (y – k)2 = r2

    • (standard equation of a circle)


Examples18
Examples

  • If the circle has a radius of 7.1 and a center at ( -4, 0), write the equation of the circle.

  • (x – h)2 + (y – k)2 = r2

  • (x – -4)2 + (y – 0)2 = 7.12

  • (x + 4)2 + y2 = 50.41


Examples19
Examples

  • The point (1, 2) is on a circle whose center is (5, -1). Write the standard equation of the circle.

  • Find the radius. (Use the distance formula)

    • .

    • .

    • .

  • (x – 5)2 + (y – -1)2 = 52

  • (x – 5)2 + (y +1)2 = 25


Graphing a circle
Graphing a Circle

  • The equation of the circle is: (x + 2)2 + (y – 3)2 = 9

  • Rewrite the equation to find the center and the radius.

  • (x – (-2))2 + (y – 3)2 = 32

  • The center is (-2, 3) and the radius is 3.


Graphing a circle1
Graphing a Circle

  • The center is (-2, 3) and the radius is 3.


Examples20
Examples

  • Do Practice 10.6C or B together.


ad