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

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


Find the measure of the blue arc or angle

Find the measure of the blue arc or angle

  • mQTS =

  • m NMP =


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.


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