# Lesson 6.1 Tangent Properties

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1 Lesson 6.1 angent roperties Name eriod ate 1. Ras r and s are tangents. w 2. is tangent to both circles and m 295. mqx r w 54 s 3. Q is tangent to two eternall tangent noncongruent circles, and N. X Q Q a. mnq, mq b. What kind of quadrilateral is NQ? plain our reasoning. N 4. is tangent to circle. Find the equation of. 5.,,, and are tangents. plain wh. (3, 9) 6. ircle has diameter 16.4 cm. ircle has diameter 6.7 cm. a. If and are internall tangent, what is the distance between their centers? b. If and are eternall tangent, what is the distance between their centers? 7. onstruct a circle,. ick a point,, on the circle. onstruct a tangent through. ick a point,, on the tangent. onstruct a second tangent to the circle through. iscovering Geometr ractice Your Skills HR 6 39

2 Lesson 6.2 hord roperties Name eriod ate In ercises 1 6, find each unknown or write cannot be determined. 1. a, b, 2. w, v 3. z c a 95 b c w 6 6 v z 4. w,, 5. w,, 6., w 66 w cm 7.. N is a 8. What s wrong with 9. Find the coordinates of. this picture? and. Justif our answer. 6 (3, 6) N 2 6 (5, 2) 10. m m m race part of a circle onto patt paper. Fold to find the center. plain our method. m HR 6 iscovering Geometr ractice Your Skills

3 Lesson 6.3 rcs and ngles Name eriod ate 1. mx 80 X 2. is a tangent. mxn mxn mn 80 Y N z z 3. a 4. a b c b a c 70 b c a 140 c b and are tangents. 6. is a tangent. is a diameter. 40 m m m 54 m 7. m 8. p m q s 29 r m r 87 m s p q 9. Find the lettered angle and arc measures. and Z are tangents. a b c d e f g h j k m n 1 2 n 25 e d c 50 m a b k Z f j h g iscovering Geometr ractice Your Skills HR 6 41

4 Lesson 6.4 roving ircle onjectures Name eriod ate In ercises 1 4, complete each proof with a paragraph or a flowchart. 1. Given: ircles and are eternall tangent, with common tangents and Show: bisects at X X 2. Given: ircle with diameter and chord.. Show: 3. Given: Q and RS are tangent to both circles. Show: Q RS. X S N R Q 4. rove the converse of the hord rcs onjecture: If two arcs in a circle are congruent, then their chords are congruent. Hint: raw radii. Given: Show: 42 HR 6 iscovering Geometr ractice Your Skills

5 Lesson 6.5 he ircumference/iameter Ratio Name eriod ate In ercises 1 4, leave our answers in terms of. 1. If r 10.5 cm, find. 2. If 25 cm, find r. 3. What is the circumference of a circle whose 4. What is the diameter of a circle whose radius is 30 cm? circumference is 24 cm? In ercises 5 9, round our answer to the nearest 0.1 unit. Use the smbol to show that our answer is an approimation. 5. If d 9.6 cm, find. 6. If 132 cm, find d and r. 7. dinner plate fits snugl in a square bo with perimeter 48 inches. What is the circumference of the plate? 8. Four saucers are part of the same set as the dinner plate in ercise 7. ach has a circumference of 15.7 inches. Will the fit, side b side, in the same square bo? If so, how man inches will there be between the saucers for padding? 9. and S are tangents. 12 cm. What is the circumference of circle? 10. How can ou use a large carpenter s square to find the circumference of a tree? S 11. In order to increase the circumference of a circle from 16 cm to 20 cm, b how much must the diameter increase? iscovering Geometr ractice Your Skills HR 6 43

6 Lesson 6.6 round the World Name eriod ate 1. lfonzo s izzeria bakes olive pieces in the outer crust of its 20-inch (diameter) pizza. here is at least one olive piece per inch of crust. How man olive pieces will ou get in one slice of pizza? ssume the pizza is cut into eight slices. 2. o use the machine at right, ou turn the crank, which turns the pulle wheel, which winds the rope and lifts the bo. hrough how man rotations must ou turn the crank to lift the bo 10 feet? 7.5 in. 10 ft 3. satellite in geostationar orbit stas over the same spot on arth. he satellite completes one orbit in the same time that arth rotates once about its ais (23.93 hours). If the satellite s orbit has radius m, calculate the satellite s orbital speed (tangential velocit) in meters per second. o 4. You want to decorate the side of a clindrical can b coloring a rectangular piece of paper and wrapping it around the can. he paper is 19 cm b 29 cm. Find the two possible diameters of the can to the nearest 0.01 cm. ssume the paper fits eactl. 5. s ou sit in our chair, ou are whirling through space with arth as it moves around the sun. If the average distance from arth to the sun is m and arth completes one revolution ever das, what is our sitting speed in space relative to the sun? Give our answer in km/h, rounded to the nearest 100 km/h. 44 HR 6 iscovering Geometr ractice Your Skills

7 Lesson 6.7 rc Length Name eriod ate In ercises 1 10, leave our answers in terms of. 1. Length of 2. he circumference is he length of F is 5. and m 60. Length Radius of F 4. Length of XY 5. he radius is 20. Length 6. he circumference is 25. of Length of 10 Y X 7. he diameter is 40. Length 8. he length of XY is Length of of iameter 110 X Y circle has an arc with measure 80 and length 88. What is the diameter of the circle? iscovering Geometr ractice Your Skills HR 6 45

8 ploration Intersecting Secants, angents, and hords Name eriod ate F is tangent to circle at point. m, m F and are tangents. m, m 4. is a tangent, m 150 m, m ,, z 6.,, z z 44 z 7. and are tangents.,, z z 8. is a tangent, m 75, HR 6 iscovering Geometr ractice Your Skills

9 4. Flowchart roof LSSN 6.1 angent roperties 1. w mqx a. mnq 90, mq 90 b. rapezoid. ossible eplanation: and NQ are both perpendicular to Q, so the are parallel to each other. he distance from to Q is, and the distance from N to Q is NQ. ut the two circles are not congruent, so NQ. herefore, N is not a constant distance from Q and the are not parallel. actl one pair of sides is parallel, so NQ is a trapezoid ossible answer: angent segments from a point to a circle are congruent. So,,, and. herefore,. 6. a cm b cm 7. pposite sides of parallelogram LSSN 6.2 hord roperties efinition of parallelogram X Y I onjecture X Y S onjecture X Y 1. a 95, b 85, c v cannot be determined, w 90 X Y oth are z w 100, 50, w 49, 122.5, cm, cannot be determined 7. Kite. ossible eplanation: N because congruent chords and are the same distance from the center. N because the are halves of congruent chords. So, N has two pairs of adjacent congruent sides and is a kite. 8. he perpendicular segment from the center of the circle bisects the chord, so the chord has length 12 units. ut the diameter of the circle is 12 units, and the chord cannot be as long as the diameter because it doesn t pass through the center of the circle. 9. (0,1), (4, 2) 10. m 49, m 253, m 156, m ossible answer: Fold and crease to match the endpoints of the arc. he crease is the perpendicular bisector of the chord connecting the endpoints. Fold and crease so that one endpoint falls on an other point on the arc. he crease is the perpendicular bisector of the chord between the two matching points. he center is the intersection of the two creases. LSSN 6.3 rcs and ngles 1. mxn 40, mxn 180, mn , 60, z a 90, b 55, c a 50, b 60, c enter 6. m 90, m 72, m 36, m m 140, m 30, m 60, m p 128, q 87, r 58, s a 50, b 50, c 80, d 50, e 130, f 90, g 50, h 50, j 90, k 40, m 80, n NSWRS iscovering Geometr ractice Your Skills

10 LSSN 6.4 roving ircle onjectures 1. Flowchart roof X X angent Segments onjecture X X ransitivit X X angent Segments onjecture 4. Flowchart roof onstruct radii,,, and. Given efinition of arc measure bisects at X efinition of segment bisector Radii of same circle Radii of same circle 2. ngles are numbered for reference. 1 SS onjecture aragraph roof It is given that, so 2 1 b the onjecture. ecause and are radii, the are congruent, so is isosceles. herefore 4 1 b the I onjecture. oth 2 and 4 are congruent to 1, so 2 4. the I onjecture, 4 3, so 2 3. he measure of an arc equals the measure of its central angle, so because their central angles are congruent,. 3. Flowchart roof X RX angent Segments onjecture X XQ Q Segment addition X XQ RX XS ddition ropert of qualit Q RS ransitivit XQ XS angent Segments onjecture RX XS RS Segment addition LSSN 6.5 he ircumference/iameter Ratio cm 2. r 12.5 cm cm 4. d 24 cm cm 6. d 42.0 cm, r 21.0 cm in. 8. Yes; about 2.0 in. 5 in. 2 in cm 10. ress the square against the tree as shown. easure the tangent segment on the square. he tangent segment is the same length as the radius. Use 2r to find the circumference. Q RS efinition of congruent segments ree cm iscovering Geometr ractice Your Skills NSWRS 105

11 LSSN 6.6 round the World 1. t least 7 olive pieces 2. bout 2.5 rotations ( ) 3085 m/s (about 3 km/s or just ( ) under 2 mi/s) cm or 9.23 cm ( Sitting speed ) ( ) 107,500 km/h LSSN 6.7 rc Length or XLRIN Intersection Secants, angents, and hords m 70, m m 114, m m 75, m , 110, z , 150, z , 68, z , ossible answers: he two points where the figure and the image intersect determine. r connect an two corresponding points and construct the perpendicular bisector, which is. N 5. 3-fold rotational smmetr, 3 lines of reflection 6. 2-fold rotational smmetr 7. 1 line of reflection 8. 1 line of reflection 9. 2-fold rotational smmetr, 2 lines of reflection fold rotational smmetr line of reflection N LSSN 7.1 ransformations and Smmetr 1. I I fold rotational smmetr, 4 lines of reflection R R 2. R L Q L R 106 NSWRS iscovering Geometr ractice Your Skills

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