10-1 Circles & Circumference
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1 10-1 Circles & Circumference Radius- Circle- Formula- Chord- Diameter- Circumference- Formula- Formula- Two circles are congruent if and only if they have congruent radii All circles are similar Concentric circles are coplanar circles that have the same center 2 points of intersection 1 point of intersection NO points of intersection
2 Example 4 Find the circumference of a helipad on the top of a hospital if the diameter of the pad is 37 meters. Example 5 Find the diameter and radius of a circle to the nearest hundredth if the circumference of the circle is millimeters. Area of a circle (part1) Areas of Circles Example 1 What is the area of the circular putting green shown to the nearest square foot? Example 2 Use the Area of a Circle to Find a Missing Measure Find the radius of a circle with an area of 46 square centimeters.
3 Sector of a circle (part2) Sectors Area of a Sector Example 1) A circular pizza has a diameter of 16 inches and is cut into 10 congruent slices. What is the area of one slice to the nearest hundredth? Find the area of each shaded sector. Round to the nearest tenth. ex 2) ex3) ex4) Ex5) Jason wants to make a spinner for a new board game he invented. The spinner is a circle divided into 8 congruent pieces, what is the area of each piece to the nearest tenth?
4 10-2 Arcs and Chords TERM: DEFINE: DRAW and LABEL Arc Central Angle Minor Arc shade minor arc: LY Semi-Circle shade semi-circle: ACB Major Arc shade major arc: LUY Arc Addition Postulate Congruent arcs 1. Using the letters shown in the diagram, name: a. four central angles,,, b. two semicircles, c. four minor arcs,,, d. four major arcs,,, 2. Using the same circle, find the measure of each arc. a. mbc b. mab c. mabc d. mcd e. mabd f. mbda
5 Example 1 Find measures of Central Angles Find m QAM. Example 2 In O, m AOB = 35, m DOE = 42, and OB OC. Identify each arc as a major arc, minor arc, or semicircle. Then find its measure. a. m AB b. mcde Example 3 Refer to the circle graph. a. Find the measurement of the central angle for each category. c. m AE b. Use the categories to identify any arcs that are congruent. Example 4 Find each measure in F. Example 5 Find the length of ZY. Round to the nearest hundredth. A B C F E D a. Y b. 6 in. 80 Z X Y 12 cm X 120 Z
6 ARC-CHORD Relationships Congruent central angles have congruent chords Congruent chords have congruent arcs 10-3 Arcs and Chords Example 1 In the figures, J K and MN PQ. Find PQ. M N 3x + 4 J P x + 12 K Q Congruent arcs have congruent central angles DIAMETER(radius)-CHORD Relationships In a circle, if a radius (or diam.) is to a chord, then it bisects the chord and its arc In a circle, the bisector of a chord is a radius (or diam.) Example 2 In S, m PQR = 86. Find m PQ. P T S Q 8 R Example 3 Find NP. Example 4 Find QR. In the same circle or in circles, 2 chords are if and only if they are equidistant from the center Example 5 In P, QR = 7x 20 and TS = 3x. What is x?
7 10-4 Inscribed Angles Inscribed Angle: Intercepted Arc: Inscribed Angle Theorem The measure of an inscribed angle is half the measure of its intercepted arc. If inscribed angles of a circle intercept the same arc, then the angles are congruent. Theorem 10.8 Theorem 10.9 An inscribed angle intercepts a semi-circle (or diam.), If a quadrilateral is inscribed in a circle, then if and only if the angle is a right angle. its opposite angles are supplementary. B A C Quadrilateral ABCD is inscribed in the circle. The circle is circumscribed. BC is a diameter. The sum of the angles of a quadrilateral is m BAC D and are supplementary. mbac C and are supplementary.
8 Find the measure of each angle or segment for each figure. 1) mps 3) madc 2) m PRU 4) m DAE 5) x= 7) x= 6) m WUT 8) m Y 9) m R 11) x 10) x 12) m B 13) mrus 14) a 15) m RUS
9 10-5 Tangents Tangent- Common tangent Thm 10.10: In a plane, a line is tangent to a circle if and only if it is to a radius drawn to the point of tangency. Thm 10.11: If 2 segment from the same exterior point are tangent to a circle then they are. Example 2 1. the center Example 1: Draw the common tangent(s) a. b. Example 3: JK is tangent to H. Find x. 2. diameter 3. a point of tangency 4. three radii,, 5. a tangent 6. two chords, Example 4: GH and KH are tangent to F. Find a. Example 5: RS and RT are tangent to Q. Find n. Example 6:Find x if ZB is tangent to Y. Ex 7: Find x and the perimeter of the triangle. In the picture, the circle is inscribed and the triangle is circumscribed.
10 10-6 Secants, Tangents, and Angle Measures Secant- Interior Angles formed by 2 intersecting chords or secants Formula: Example 1 Find x. a. R T A secant & a tangent intersect at a point of tangency Formula: 62 S V x U 144 Example 2. Find each measure a. m QPR Q 136 R b.m F P S E 82 C D b. A 135 E C B x 53 D T c. G x K L 120 H 92 J
11 Vertex of Angle PICTURE Angle Measure On the circle Inside the circle Outside the circle ex5) x= ex6) x= ex7) x=
12 CHORD-CHORD THM 10-7 Special Segments in a Circle Formula: SECANT-SECANT-THM Formula: TANGENT-SECANT THM Formula: Find x in each picture. ex1) ex2) ex3) ex4) Find x in each picture. ex5) ex6)
13 Find x in each picture. ex7) ex 8) ex9) ex10) ex11) ex12)
14 10-8 Equations of Circles Equation of a Circle in Standard Form: Example 1 Write the equation of each circle. a. center at (1, -5), d = 6 b. center at (-6, 0), r = 8 Example 2 Write the equation of the circle with center at ( 3, 5) that passes through (3, 3). Example 3 The equation of a circle is x 2 + y 2 2x 10y = 22. State the coordinates of the center and the measure of the radius. Then graph the equation. Example 5 Intersections with Circles Find the point(s) of intersection between x 2 + y 2 = 20 and y = 2x.
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