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

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

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

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

Q

P

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.

Q

P

Tangent Theorems

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

QP

Examples

Find an example for each term:Center

Chord

Diameter

Radius

Point of Tangency

Common external tangent

Common internal tangent

Secant

Examples

The diameter is given. Find the radius.

d=15cmd=6.7ind=3ftd=8cm

Examples

The radius is given. Find the diameter.

r = 26inr = 62ftr = 8.7inr = 4.4cm

Examples

Tell whether AB is tangent to C.

A

B

C

14

5

15

Examples

Tell whether AB is tangent to C.

A

B

C

16

12

8

Examples

AB and AD are tangent to C. Find x.

A

B

D

2x + 7

5x - 8

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

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.

Examples

Find the measure of each arc.

MNMPNPMN

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

Examples

Find the measure of each arc.

GE

GEF

GF

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.

Examples

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

Examples

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

mMN

mMKN

mJML

Examples

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

Homework

Pg. 600 # 26-28, 37, 39, 47, 48Pg. 607 # 12-30 even, 32-34, 37-38

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

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

mADB = ½ mAB

Find the measure of the blue arc or angle

mQTS =

m NMP =

Theorem

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

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.

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.

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 m  D + m  F = 180° and m  E + m  G = 180°

Examples

Find the value of each variable.

More Examples

Pg 616 # 2-8

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

Examples

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

Examples

BC is tangent to the circle. Find m CBD.

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

Theorems

m 1 = ½ (mBC – mAC)

m 2 = ½ (mPQR – mPR)

m 3 = ½ (mXY – mWZ)

1

C

BA

2

P

R

Q

3

WX

YZ

More Examples

Pg. 624 #2-7

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.

Examples

Find x.

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.

Theorems

EA EB = EC ED

E

A

C

B

D

Theorems

(EA)2 = EC EDEA is a tangent segment, ED is a secant

segment.

E

A

CD

Examples

Find x.

Examples

Find x.

Examples

x ___ = 10 ___ X2 = 4 ____

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)

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

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

22 )21()15( r22 )3(4 r

5r

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 Circle

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

Examples

Do Practice 10.6C or B together.