Circles
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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.

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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.

P

Q


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 Theorems

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

Q

P


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=15cm

    • d=6.7in

    • d=3ft

    • d=8cm


Examples

  • The radius is given. Find the diameter.

    • r = 26in

    • r = 62ft

    • r = 8.7in

    • r = 4.4cm


Examples

  • Tell whether AB is tangent to C.

A

14

5

B

15

C


Examples

  • Tell whether AB is tangent to C.

A

12

C

16

8

B


Examples

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

D

2x + 7

A

5x - 8

B


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.

    • MN

    • MPN

    • PMN


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.

    • PQ

    • SU

    • QT

    • TUP

    • PUQ


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, 48

  • Pg. 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

mD + mF = 180° and mE + mG = 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

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 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

B

A

E

C

D


Theorems

  • (EA)2 = EC ED

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

A

E

C

D


Examples

  • Find x.


Examples

  • Find x.


x ___ = 10 ___

X2 = 4 ____

Examples


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


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.


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