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Circles Chapter 10

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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.

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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.

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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.

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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

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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

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

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Examples Find an example for each term: Center Chord Diameter Radius Point of Tangency Common external tangent Common internal tangent Secant

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Examples The diameter is given. Find the radius. d=15cm d=6.7in d=3ft d=8cm

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Examples The radius is given. Find the diameter. r = 26in r = 62ft r = 8.7in r = 4.4cm

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Examples Tell whether AB is tangent to C. A B C

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Examples Tell whether AB is tangent to C. A B C

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Examples AB and AD are tangent to C. Find x. A B D 2x + 7 5x - 8

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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.

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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.

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Examples Find the measure of each arc. MN MPN PMN

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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

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Examples Find the measure of each arc. GE GEF GF

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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.

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Examples Determine whether the arc is minor, major or a semicircle. PQ SU QT TUP PUQ

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Examples KN and JL are diameters. Find the indicated measures. mKL mMN mMKN mJML

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

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

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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.

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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

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Find the measure of the blue arc or angle mQTS = m NMP =

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

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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.

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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.

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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°

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Examples Find the value of each variable.

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More Examples Pg 616 # 2-8

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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

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

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

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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 ( ) x = 140

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Theorems m 1 = ½ (mBC – mAC) m 2 = ½ (mPQR – mPR) m 3 = ½ (mXY – mWZ) 1 C B A 2 P R Q 3 W X Y Z

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More Examples Pg. 624 #2-7

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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.

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Examples Find x.

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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.

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Theorems EA EB = EC ED E A C B D

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Theorems (EA) 2 = EC ED EA is a tangent segment, ED is a secant segment. E A C D

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Examples Find x.

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Examples Find x.

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Examples x ___ = 10 ___ X 2 = 4 ____

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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 = r 2 (standard equation of a circle)

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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 = r 2 (x – -4) 2 + (y – 0) 2 = (x + 4) 2 + y 2 = 50.41

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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 = 5 2 (x – 5) 2 + (y +1) 2 = 25

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

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

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Examples Do Practice 10.6C or B together.

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