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# Circles - PowerPoint PPT Presentation

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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## PowerPoint Slideshow about 'Circles' - jett

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### Circles

Chapter 10

• 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.

• 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.

• 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.

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

P

Q

• 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

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

Q

P

• Find an example for each term:

Center

Chord

Diameter

Point of Tangency

Common external tangent

Common internal tangent

Secant

• The diameter is given. Find the radius.

• d=15cm

• d=6.7in

• d=3ft

• d=8cm

• The radius is given. Find the diameter.

• r = 26in

• r = 62ft

• r = 8.7in

• r = 4.4cm

• Tell whether AB is tangent to C.

A

14

5

B

15

C

• Tell whether AB is tangent to C.

A

12

C

16

8

B

• AB and AD are tangent to C. Find x.

D

2x + 7

A

5x - 8

B

• 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.

• 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.

• Find the measure of each arc.

• MN

• MPN

• PMN

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

• mABC = mAB + mBC

• Find the measure of each arc.

• GE

• GEF

• GF

• 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.

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

• PQ

• SU

• QT

• TUP

• PUQ

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

• mKL

• mMN

• mMKN

• mJML

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

• Pg. 600 # 26-28, 37, 39, 47, 48

• Pg. 607 # 12-30 even, 32-34, 37-38

• 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.

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

• mQTS =

• m NMP =

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

• 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.

• 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.

• 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°

• Find the value of each variable.

• Pg 616 # 2-8

• 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

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

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

• 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

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

• Pg. 624 #2-7

• 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.

• Find x.

• 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.

• EA EB = EC ED

B

A

E

C

D

• (EA)2 = EC ED

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

A

E

C

D

• Find x.

• Find x.

X2 = 4 ____

Examples

• 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)

• 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

• 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

• 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.

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

• Do Practice 10.6C or B together.