Chapter Review Lines that Intersect Circles Arcs and Chords. Identify each line or segment that intersects each circle.
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1 HPTR 11-1 hapter Review 11-1 Lines that Intersect ircles Identify each line or segment that intersects each circle. 1. m 2. N L K J n W Y X Z V 3. The summit of Mt. McKinley in laska is about 20,321 feet above sea level. What is the distance from the summit to the horizon, to the nearest mile? (Hint: 5280 ft 1 mile, radius of the arth 4000 miles) 11-2 rcs and hords ind each measurement G 7. G opyright by Holt, Rinehart and Winston. 253 Geometry
2 HPTR 11-1 hapter Review 11-1 Lines that Intersect ircles Identify each line or segment that intersects each circle. 1. m 2. N L K J n W Y X Z V radii: KN,KJ,KL tangent: m chords: JL secant: JK diameter: JL radii: XV,XW tangent: n chords: ZY,WV secant: YZ diameter: WV 3. The summit of Mt. McKinley in laska is about 20,321 feet above sea level. What is the distance from the summit to the horizon, to the nearest mile? (Hint: 5280 ft 1 mile, radius of the arth 4000 miles) miles 11-2 rcs and hords ind each measurement G 7. G opyright by Holt, Rinehart and Winston. 253 Geometry
3 HPTR 11 RVIW ONTINU ind each length to the nearest tenth JK J 5 L 8 K M 11-3 Sector rea and rc Length ind the shaded area. Round to the nearest tenth, if necessary. 10. s part of a parks beautification committee Kelly is designing a circular flower garden. She plans to divide the flower garden into sectors and plant different colored flowers in each sector. What is the area of each sector to the nearest square foot? 70 6 ind each arc length. Give your answer in terms of and rounded to the nearest hundredth G 65 G cm 13. an arc with measure 54 in 14. a semicircle in a circle with diameter a circle with diameter 14 in. 112 m opyright by Holt, Rinehart and Winston. 254 Geometry
4 HPTR 11 RVIW ONTINU ind each length to the nearest tenth JK L 8 J K M 11-3 Sector rea and rc Length ind the shaded area. Round to the nearest tenth, if necessary. 10. s part of a parks beautification committee Kelly is designing a circular flower garden. She plans to divide the flower garden into sectors and plant different colored flowers in each sector. What is the area of each sector to the nearest square foot? 22 ft ind each arc length. Give your answer in terms of and rounded to the nearest hundredth G G 3.2 cm 13. an arc with measure 54 in 14. a semicircle in a circle with diameter a circle with diameter 14 in. 112 m opyright by Holt, Rinehart and Winston. 254 Geometry
5 HPTR 11 RVIW ONTINU 11-4 Inscribed ngles ind each measure. 15. m 16. m or Problems 15 and or Problems 17 and m 18. m ngle Relationships in ircles ind each measure. 19. m LMN 20. m L M N 21. ind the measure of angle x. Q P x 150 R opyright by Holt, Rinehart and Winston. 255 Geometry
6 HPTR 11 RVIW ONTINU 11-4 Inscribed ngles ind each measure. 15. m 16. m or Problems 15 and or Problems 17 and m 18. m ngle Relationships in ircles ind each measure. 19. m LMN 20. m L M N 21. ind the measure of angle x. Q 30 P x 150 R opyright by Holt, Rinehart and Winston. 255 Geometry
7 LSSON 11-6 ONTINU 11-6 Segment Relationships in ircles ind the value of the variable and the length of each chord or secant segment x J x 24. section of an arched bridge is based on an arc of a circle as shown. is the perpendicular bisector of. 60 ft, and 20 ft. What is the diameter of the circle? 11-7 ircles in the oordinate Plane Write the equation of each circle. 25. with center (3, 4) and 26. that passes through ( 4, 12) radius 5 has center ( 12, 6) 27. local technology company is planning the location of a new cell phone tower to help with coverage in three cities. To optimize cell phone coverage, the tower should be equidistant from the three cities which are located on a coordinate plane at (4, 10), (6, 4), and ( 10, 4). What are the coordinates where the tower should be built? opyright by Holt, Rinehart and Winston. 256 Geometry
8 LSSON 11-6 ONTINU 11-6 Segment Relationships in ircles ind the value of the variable and the length of each chord or secant segment x J x x 8, 14, 16 x 32.2, 43.2; section of an arched bridge is based on an arc of a circle as shown. is the perpendicular bisector of. 60 ft, and 20 ft. What is the diameter of the circle? 65 ft 11-7 ircles in the oordinate Plane Write the equation of each circle. 25. with center (3, 4) and 26. that passes through ( 4, 12) radius 5 has center ( 12, 6) (x 3) 2 (y 4) (x 12) 2 (y 6) local technology company is planning the location of a new cell phone tower to help with coverage in three cities. To optimize cell phone coverage, the tower should be equidistant from the three cities which are located on a coordinate plane at (4, 10), (6, 4), and ( 10, 4). What are the coordinates where the tower should be built? ( 2, 2) opyright by Holt, Rinehart and Winston. 256 Geometry
9 HPTR 11 Postulates and Theorems Theorem Theorem Theorem Postulate Theorem Theorem Theorem Theorem orollary Theorem Theorem Theorem Theorem If a line is tangent to a circle, then it is perpendicular to the radius drawn to the point of tangency. (line tangent to line to radius) If a line is perpendicular to a radius of a circle at a point on the circle, then the line is tangent to the circle. (line to radius line tangent to ) If two segments are tangent to a circle, from the same external point, then the segments are congruent. (2 segs. tangent to from same ext. pt. segs ) (rc ddition Postulate) The measure of an arc formed by two adjacent arcs is the sum of the measure of the two arcs. In a circle or congruent circles: (1) ongruent central angles have congruent chords. (2) ongruent chords have congruent arcs. (3) ongruent arcs have congruent central angles. In a circle, if a radius (or diameter) is perpendicular to a chord, then it bisects the chord and its arc. In a circle, the perpendicular bisector of a chord is a radius (or a diameter). The measure of an inscribed angle is half the measure of its intercepted arc. If inscribed angles of a circle intercept the same arc or are subtended by the same chord or arc, then the angles are congruent. n inscribed angle subtends a semicircle if and only if the angle is a right angle. If a quadrilateral is inscribed in a circle, then its opposite angles are supplementary. If a tangent and a secant (or chord) intersect on a circle at the point of tangency, then the measure of the angle formed is half the measure of its intercepted arc. If two secants or chords intersect in the interior of a circle, then the measure of each angle formed is half the sum of the measures of its intercepted arcs. opyright by Holt, Rinehart and Winston. 257 Geometry
10 HPTR 11 Postulates and Theorems Theorem Theorem Theorem Postulate Theorem Theorem Theorem Theorem orollary Theorem Theorem Theorem Theorem If a line is tangent to a circle, then it is perpendicular to the radius drawn to the point of tangency. (line tangent to line to radius) If a line is perpendicular to a radius of a circle at a point on the circle, then the line is tangent to the circle. (line to radius line tangent to ) If two segments are tangent to a circle, from the same external point, then the segments are congruent. (2 segs. tangent to from same ext. pt. segs ) (rc ddition Postulate) The measure of an arc formed by two adjacent arcs is the sum of the measure of the two arcs. In a circle or congruent circles: (1) ongruent central angles have congruent chords. (2) ongruent chords have congruent arcs. (3) ongruent arcs have congruent central angles. In a circle, if a radius (or diameter) is perpendicular to a chord, then it bisects the chord and its arc. In a circle, the perpendicular bisector of a chord is a radius (or a diameter). The measure of an inscribed angle is half the measure of its intercepted arc. If inscribed angles of a circle intercept the same arc or are subtended by the same chord or arc, then the angles are congruent. n inscribed angle subtends a semicircle if and only if the angle is a right angle. If a quadrilateral is inscribed in a circle, then its opposite angles are supplementary. If a tangent and a secant (or chord) intersect on a circle at the point of tangency, then the measure of the angle formed is half the measure of its intercepted arc. If two secants or chords intersect in the interior of a circle, then the measure of each angle formed is half the sum of the measures of its intercepted arcs. opyright by Holt, Rinehart and Winston. 257 Geometry
11 POSTULTS N THORMS ONTINU Theorem Theorem Theorem Theorem Theorem 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 half the difference of the measures of its intercepted arcs. (hord-hord Product Theorem) If two chords intersect in the interior of a circle, then the products of the lengths of the segments of the chords are equal. (Secant-Secant Product Theorem) If two secants intersect in the exterior of a circle, then the product of the lengths of one secant segment and its external segment equals the product of the lengths of the other secant segment and its external segment. (whole outside whole outside) (Secant-Tangent Product Theorem) If a secant and a tangent intersect in the exterior of a circle, then the product of the lengths of the secant segment and its external segment equals the length of the tangent segment squared. (whole outside tangent 2 ) (quation of a ircle) The equation of a circle with center (h, k) and radius r is (x h) 2 (y k) 2 r 2. opyright by Holt, Rinehart and Winston. 258 Geometry
12 POSTULTS N THORMS ONTINU Theorem Theorem Theorem Theorem Theorem 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 half the difference of the measures of its intercepted arcs. (hord-hord Product Theorem) If two chords intersect in the interior of a circle, then the products of the lengths of the segments of the chords are equal. (Secant-Secant Product Theorem) If two secants intersect in the exterior of a circle, then the product of the lengths of one secant segment and its external segment equals the product of the lengths of the other secant segment and its external segment. (whole outside whole outside) (Secant-Tangent Product Theorem) If a secant and a tangent intersect in the exterior of a circle, then the product of the lengths of the secant segment and its external segment equals the length of the tangent segment squared. (whole outside tangent 2 ) (quation of a ircle) The equation of a circle with center (h, k) and radius r is (x h) 2 (y k) 2 r 2. opyright by Holt, Rinehart and Winston. 258 Geometry
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