C=2πr C=πd. Chapter 10 Circles Circles and Circumference. Circumference: the distance around the circle

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1 10.1 Circles and Circumference Chapter 10 Circles Circle the locus or set of all points in a plane that are A equidistant from a given point, called the center When naming a circle you always name it by its center. Circle A Radius the distance from the center to a point on the circle The radius is half of the diameter. Diameter the distance across a circle, through the center The diameter is twice the radius. Chord a segment whose endpoints are points on the circle Concentric circles coplanar circles that have a common center Example: M Circle all of the segments that are chords. MN MO LM LN L O N Name the circle: Name a radius: Name a diameter: Finding Measures in Intersecting Circles Example: The diameter of Circle S is 26 units, the radius of Circle R is 14 units, and DS = 5 units. Find CD. Circumference: the distance around the circle C=2πr C=πd Example: Find the circumference of circle A and circle B. A B

2 Example: If the circumference of a circle is 65.4 feet find the diameter and radius. Round to the nearest hundredth. Area: the amount of space inside the circle A=πr 2 Example: Find the area of circle A and circle B. A B Example: If the area of a circle is 256 feet, find the diameter. Round to the nearest hundredth. Inscribed Circle the largest possible circle that can be drawn inside a polygon Each side of the polygon is tangent. Circumscribed Circle the circle that can be drawn outside a polygon passing through all the vertices of the polygon

3 10.2 Measuring Arcs and Angles Central angle an angle whose vertex is the center of a circle Central Angle = Measure of Arc Arc measure the same as its corresponding central angle Sum of Central Angles the sum of the central angles in a circle is 360 Minor arc part of a circle that measures less than 180 Smaller than semicircle. < 180 Major arc part of a circle that measures between 180 and 360 Smaller than semicircle. > 180 Semicircle an arc whose endpoints are the endpoints of a diameter of the circle = 180 A semicircle measures exactly 180. Arc Addition Postulate: The measure of an arc formed by two adjacent arcs is the sum of the measures of the two arcs. AD DB ADB Theorem 10.1: In the same circle, or in congruent circles, two minor arcs are congruent if and only if their corresponding chords are congruent. AD BC Example: In circle E, find the measure of the angle or the arc named. BC AC ADB 1 2

4 Example: In circle C with diameter SP, find the measure of each angle and classify each arc as a major arc, minor arc, or semicircle then find its measure. ST PCQ SQP SCQ SQ SCP SPQ TCP SPT TSQ PT Arc Length: the distance between the endpoints of an arc l x r Example: Find the length of the PQ. Round to the nearest hundredth. Example: Find the length of the BDC. Round to the nearest hundredth. 4

5 10.3 Arcs and Chords Theorem 10.2: In the same circle, or in congruent circles, two minor arcs are congruent if and only if their corresponding chords are congruent. AB AB CD CD Example: Find the value of x in each circle. Theorem 10.3: If a diameter of a circle is perpendicular to a chord, then the diameter bisects the chord and its arc. TU UR TS RS Theorem 10.4: If one chord is a perpendicular bisector of another chord, then the first chord is a diameter. QS is the diameter Example: Find the value of x in each circle. a. b. Theorem 10.5: In the same circle, or in congruent circles, two chords are congruent if and only if they are equidistant from the center. Example: Find the value of x in each circle. a. b.

6 10.4 Inscribed Angles Inscribed angle an angle whose vertex is on a circle and whose sides contain chords of the circle Intercepted arc the arc that lies in the interior of an inscribed angle and has endpoints on the angle. ] Theorem 10.6: If an angle is inscribed in a circle, then its measure is half the measure of its intercepted arc. Example: Find x. Measure of an Inscribed Angle: Example: Find DC and ADB Example: Find the values of x, y, and z. Theorem 10.7 : If two inscribed angles of a circle intercept the same arc, then the angles are congruent. A C A B D Example: Find x. B

7 Inscribed polygon drawn inside. A polygon is inscribed in a circle if all its vertices lie on the circle. Theorem 10.8 : If a right triangle is inscribed is inscribed in a circle, then the hypotenuse is a diameter of the circle. Conversely, if one side of an inscribed triangle is a diameter of the circle, then the triangle is a right triangle and the angle opposite the diameter is the right angle. Theorem 10.9 : A quadrilateral can be inscribed in a circle if and only if its opposite angles are supplementary. Example: Find the value of x and y.

8 10.5 Tangents Tangent a line in the plane of a circle that intersects the circle in exactly point. Point of Tangency the point at which the tangent line intersects the circle. 2 TYPES: Common External Tangents Common Internal Tangents Theorem : If a line is tangent to a circle, then it is perpendicular to the radius drawn to the point of tangency. Theorem : If two segments from the same exterior point are tangent to a circle, then they are congruent.

9 Example: Find the x then find the perimeter of the polygon Secants, Tangents, & Angle Measures Secant a line that intersects a circle in points. Theorem 10.12: 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. a. b. Theorem 10.13: 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. x

10 Theorem 10.14: If a tangent and a secant, two tangents, or two secants intersect in the exterior of a circle, then the measure of the angle formed is one half the difference of the measure of the intercepted arcs. a. b. a. b.

11 10.7 Special Segments in a Circle Theorem 10.15: If two chords intersect in the interior of a circle, then the product of the lengths of the segments of one chord is equal to the products of the lengths of the segments of the other chord. Theorem 10.16: If two secant segments share the same endpoint outside a circle, then the product of the length of one secant segment and the length of its external segment equals the product of the length of the other secant segment and the length of its external segment. a. b. Theorem 10.17: If a secant segment and a tangent segment share an endpoint outside a circle, then the product of the length of the secant segment and the length of its external segment equals the square of the length of the tangent segment. Example: Find the value of y.

12 10.8 Equations of Circle Standard Form ( x h) ( y k) r Center = (h, k) Radius = r Example: Find the equation of the circle with a center at (1, -8) and a radius of 7 and graph. Example: Find the equation of the circle with a center at (-3, 6) that passes through the point (0, 6) and graph.

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