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UNIT 7: CIRCLES Circle Basics Definition A circle is The radius is_________________________________________________ Other terms Name: Congruent Circles have the same Concentric circles . Internally Tangent Circles Area and Circumference of a Circle Externally Tangent Circles UNIT 7: CIRCLES Tangent to a Circle Theorem: Example 1: In A, BC is tangent at point B. a. Find AC. Round to the hundredth. b. Find DC. Round to the hundredth. Example 2: Using the triangle, determine if BC is tangent to the circle at point B. Example 3: CD is tangent to A and B at points D and C, respectively. Find the distance between the centers of the two circles. Round to the tenth. Ice Cream Cone Theorem Example 4: AB , BC , and AC are tangent to at points E, F, and D, respectively. Find the perimeter of ABC. G Circumscribed Angle Theorem Example 6: In Example 5, if mB 42 , find mADC . Example 5: Find the value of x. UNIT 7: CIRCLES Circles on the Coordinate Plane The equation of a circle Recall that a circle is the set of all points that are a fixed distance (radius) from fixed point (center). Using the distance formula, r x h y k 2 2 By squaring both sides, this gives us the general equation of a circle on the coordinate plane. r 2 x h y k 2 2 where r is the radius of the circle, and (h, k) is the center. Example 7: Write the equation of the circle with a center (-2,5) and radius of 4 units. x h y k r 2 2 2 x -2 y 5 42 2 2 x 2 y 5 16 2 2 Example 8: If the equation of a circle is x 1 y 6 10 , what is the center and radius? 2 2 x h y k r 2 2 2 x -1 y 6 10 2 2 Center (-1,6) and Radius = 10 units Example 9: If the center of a circle is (2,3) and the point (-4,-1) is on the circle, write the equation of the circle. 62 42 = Distance between center and point on circle is the radius: 52 x h y k r 2 2 2 x 2 y 3 52 2 2 Example 10: If the equation of a circle is x 1 y 6 45 , is the point 7,3 on the circle? Support your answer. 2 2 Plug in point (-7,3) for (x, y) in the equation: -7 1 3 6 45 2 2 6 3 45 2 2 36 9 45 45 45 Because it makes equation true, the point 7,3 is on the circle. UNIT 7: CIRCLES Proving that Two Circles are Similar Two figures are similar if and only if there is a sequence of similarity transformations that maps one figure onto the other. Example 11: Prove that the circles are similar by identifying a sequence of similarity transformations that will map one circle onto the other. Angles and Arc Measure Example 12: Find the following measures in A . BE is a diameter. 1. mCD = 2. mDEF = 3. mDFC = 4. mEAD = 5. Are there semicircles in the figure? If so, name them. Congruent Circle Parts Theorem Example 13: In A , mBD 125 , what is mCD ? UNIT 7: CIRCLES Example 14: C J and mDCG mNJM . Find DG. Radius to a Chord Theorem If a radius (or diameter) is perpendicular to a chord, it bisects the chord and its intercepted arc. If a line is perpendicular to and bisects a chord, then it contains the radius (or diameter) of the circle. Example 15: In R , RS 8 , and SM 9 . Find the length of the radius of the circle. Find SN and NP. Find mNP rounded to the nearest tenth. Inscribed Angles An inscribed angle is an angle with . Inscribed Angle Theorem An inscribed angle is the measure of its intercepted arc. UNIT 7: CIRCLES Semicircle Corollary An inscribed angle intercepting a semi-circle is Example 16: In . O , PM is a diameter. Find the following. mPN = mMN = mLNP = mMPN = mLPN = mMNP = mLM = mPMN = Inscribed Quadrilateral Theorem An inscribed polygon is a polygon placed inside a circle, so that A quadrilateral inscribed in a circle has opposite angles that are Example 17: Each quadrilateral is inscribed in the circle. Find the value of the variables. Example 18: Quadrilateral ABCD is inscribed in the circle. Find the all of the angle measures of quadrilateral ABCD. .