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Name: ____________________________ Chapter 10 Notes Lesson 10-1 Center: the middle Radius: A line segment drawn from the center to any point on the circle Diameter: A line segment drawn across a circle that passes through the center (twice the radius) Chord: Any segment with endpoints that are on the circle Circumference: The distance around the outside of the circle; C = 2πr, C = πd Lesson 10.2 Angles and Arcs • The sum of the measures of the central angles of a circle with no interior points is 360. • Minor arc- arc whose central angle is less than 180 • Major arc- arc whose central angle is greater than 180 • Semicircle- half a circle, 180 • Arc addition postulate- the measure of the arc formed by 2 adjacent arcs is the sum of the measures of the 2 arcs. A l 360 2r Example 1: Refer to circle T a. Find mRTS b. Find mQTR Your Turn: Refer to circle Z. a. Find mCZD b. Find mBZC 1 Example #2: Use circle P where mNPM 46, PL bisects KPM , and OP KN. a. Find mOK . b. Find mLM . c. Find mJKO Your Turn: In circle B, XP and YN are diameters, mXBN 108, and BZ bisects YBP. Find each measure. a. mYZ b. mXY c. mXNZ Example #3: BICYCLES This graph shows the percent of each type of bicycle sold in the United States in 2001. a. Find the measurement of the central angle representing each category. List them from least to greatest. b. Is the arc for the wedge named Youth congruent to the arc for the combined wedges named Other and Comfort? Your Turn: SPEED LIMITS This graph shows the percent of U.S. states that have each speed limit on their interstate highways. a. Find the measurement of the central angles representing each category. List them from least to greatest. b. Is the arc for the wedge for 65 mph congruent to the combined arcs for the wedges for 55 mph and 70 mph? 2 Example #4: In circle B, AC = 9 and mABD 40 . Find the length of AD. Your Turn: In circle A, AY = 21 and mXAY 45 . Find the length of WX. Lesson 10.3 Arcs and Chords Theorem 10.2- In a circle or in congruent circles, two minor arcs are congruent if and only if their corresponding chords are congruent Theorem 10.3- In a circle, if a diameter (or radius) is perpendicular to a chord, then it bisects the chord and its arc. Theorem 10.4- In a circle or in congruent circles, two chords are congruent if and only if they are equidistant from the center. Example #2: TESSELLATIONS The rotations of a tessellation can create twelve congruent central angles. Determine whether PQ ST . . Your Turn: ADVERTISING A logo for an advertising campaign is a pentagon that has five congruent central angles. Determine whether PTS PAS. 3 Example 3: Circle W has a radius of 10 centimeters. Radius WL is perpendicular to chord HK which is 16 centimeters long. a. If mHL = 53, find mMK. b. Find JL. Your Turn: Circle O has a radius of 25 units. Radius OC is perpendicular to the chord AE , which is 40 units long. a. If mMG = 35, find mCG. b. Find CH. Example #4: Chords EF and GH are equidistant from the center. If the radius of circle P is 15 and EF = 24, find PR and RH. Your Turn: Chords SZ and UV are equidistant from the center of circle X. If TX = 39 and XY = 15, find WZ and UV. Lesson 10.4 Inscribed Angles Theorem 10.5 Inscribed Angle Theorem If an angle is inscribed in a circle, then the measure of the angle equals one-half the measure of its intercepted arc. Theorem 10.6 If two inscribed angles of a circle (or congruent circles) intercept congruent arcs or the same arc, then the angles are congruent Theorem 10.7 If an inscribed angle intercepts a semicircle, the angle is a right angle Theorem 10.8 If a quadrilateral is inscribed in a circle, then its opposite angles are supplementary. 4 Example #1: In circle F, mWX = 20, mXY = 40, mUZ = 108, and mUW = mYZ. Find the measures of the numbered angles. Your Turn: In circle A, mXY = 60, mYZ = 80 and mWX = mWZ. Find the measures of the numbered angles. Example #3: PROBABILITY Points M and N are on a circle so that MN = 72. Suppose point L is randomly located on the same circle so that it does not coincide with M or N. What is the probability that mMLN 144 ? Your Turn: PROBABILITY Points A and X are on a circle so that mAX = 84. Suppose point B is randomly located on the same circle so that it does not coincide with A or X. What is the probability that mABX 42 ? Example #4: ALGEBRA Triangles TVU and TSU are inscribed in circle P with VU = SU. Find the measure of each numbered angle if m2 x 9 and m4 2x 6 . Your Turn: ALGEBRA Triangles MNO and MPO are inscribed in circle D with MN = OP. Find the measure of each numbered angle if m2 4x 8 and m3 3x 9 Example #5: Quadrilateral QRST is inscribed in circle M. If mQ 87 and mR 102 Find mS and mT . 5 Your Turn: Quadrilateral BCDE is inscribed in circle X. If mB 99 and mC 76 Find mD and mE . Lesson 10.5 Tangents • Theorem 10.9 If a line is tangent to a circle, then it is perpendicular to the radius drawn at the point of tangency • Theorem 10.10 If a line is perpendicular to the radius of a circle at its endpoint on the circle, then the line is tangent to the circle. • Theorem 10.11 If two segments from the same exterior point are tangent to a circle, then they are congruent. Example #1: Algebra RS is tangent to circle Q at point R. Find y. Your Turn: CD is a tangent to circle B at point D. Find a. Example #2a: Determine whether BC is tangent to circle A. to circle D. 6 Example 2b: Determine whether WE is tangent Your Turn: Determine whether ED is tangent to circle Q. to circle V. Determine whether XW is tangent Example #3: ALGEBRA Find x. Assume that segments that appear tangent to circles are tangent. Your Turn: ALGEBRA Find a. Assume that segments that appear tangent to circles are tangent. Example #4: Triangle HJK is circumscribed about circle G. Find the perimeter of HJK if NK JL 29 Your Turn: Triangle NOT is circumscribed about circle M. Find the perimeter of NOT if CT NC 28 7 Lesson 10.6 Secants, Tangents, and Angle Measures • Theorem 10.12 If two secants intersect in the interior of a circle, then the measure of an angle formed is one-half the sum of its measure of the arcs intercepted by the angle and its vertical angle. • Theorem 10.13 If a secant and tangent intersect at a point of tangency, then the measure of each angle formed is one-half the measure of the intercepted arc. • Theorem 10.14 If two secants, a secant and a tangent, or two tangents intersect in the exterior of a circle, then the measure of the angle formed is one-half the positive difference of the measures of the intercepted arcs. Ex.#1: Find m4 if mFG 88 and mEH 76. Your Turn: Find m5 if mAC 63 and mXY 21. Example #2: Find RPS if mPT = 114 and mTS = 136. Your Turn: Find FEG if mHF = 80 and mHE = 164. 8 Example #3: Find x. Your Turn: Find x. Example #4: JEWELRY A jeweler wants to craft a pendant with the shape shown. Use the figure to determine the measure of the arc at the bottom of the pendant. Your Turn: PARKS Two sides of a fence to be built around a circular garden in a park are shown. Use the figure to determine the measure of A. Example #5: Find x. Your Turn: Find x. 9 Lesson 10.7 • Theorem 10.5 If two chords intersect in a circle, then the products of the measures of the segments of the chords are equal. • Theorem 10.16 If two secant segments are drawn to a circle from an exterior point, then the product of the measures of one secant segment and its external secant segment is equal to the product of the measures of the other secant segment and its external secant segment. • Theorem 10.17 If a tangent segment and a secant segment are drawn to a circle from an exterior point, then the square of the measure of the tangent segment is equal to the product of the measures of the secant segment and its external secant segment. Example #1: Find x. Your Turn: Find x. Example #2: BIOLOGY Biologists often examine organisms under microscopes. The circle represents the field of view under the microscope with a diameter of 2 mm. Determine the length of the organism if it is located 0.25 mm from the bottom of the field of view. Round to the nearest hundredth. Your Turn: ARCHITECTURE Phil is installing a new window in an addition for a client’s home. The window is a rectangle with an arched top called an eyebrow. The diagram below shows the dimensions of the window. What is the radius of the circle containing the arc if the eyebrow portion of the window is not a semicircle? 10 Example #3: Find x if EF = 10, EH = 8, and FG = 24. Your Turn: Find x if GO = 27, OM = 25, and IK = 24. Example #4: Find x. Assume that segments that appear to be tangent are tangent. Your Turn: Find x. Assume that segments that appear to be tangent are tangent. Lesson 10.8 An equation for a circle with a center at (h, k) and a radius of r( xunits h) 2 is: ( y k )2 r 2 Example #1a: Write an equation for a circle with the center at (3, –3), d = 12. Example #1b: Write an equation for a circle with the center at (–12, –1), r = 8. Your Turn: Write an equation for each circle. a. center at (0, –5), d = 18 11 b. center at (7, 0), r =20 Example #2: A circle with a diameter of 10 has its center in the first quadrant. The lines y = –3 and x = –1 are tangent to the circle. Write an equation of the circle. Your Turn: A circle with a diameter of 8 has its center in the second quadrant. The lines y = –1 and x = 1 are tangent to the circle. Write an equation of the circle. Example #3a: Graph x 2 ( y 3) 2 4 y 2 Example #3b: x 3 y 2 16 2 y x x y Your Turn: a. Graph x 2 ( y 5) 2 25 y b. Graph x 4 ( y 3) 2 9 2 x Example #4: ELECTRICITY Strategically located substations are extremely important in the transmission and distribution of a power company’s electric supply. Suppose three substations are modeled by the points D(3, 6), E(–1, 0), and F(3, –4). Determine the location of a town equidistant from all three substations, and write an equation for the circle. Your Turn: AMUSEMENT PARKS The designer of an amusement park wants to place a food court equidistant from the roller coaster located at (4, 1), the Ferris wheel located at (0, 1), and the boat ride located at (4, –3). Determine the location for the food court and write an equation for the circle. 12 x