7 or 1.17 as your ratio of the lengths when

Transcription

1 .5. What id You Learn? ore Vocabular directed line segment, p. 50 ore oncepts Section.5 Side-Side-Side (SSS) Similarit heorem, p. 9 Side-ngle-Side (SS) Similarit heorem, p. 9 Section. riangle Proportionalit heorem, p. 500 onverse of the riangle Proportionalit heorem, p. 500 hree Parallel Lines heorem, p. 501 riangle ngle isector heorem, p. 50 Partitioning a irected Line Segment, p. 503 athematical Practices 1. In ercise 1 on page 9, wh do ou onl need to find two ratios instead of three to compare the side lengths of a pair of triangles?. In ercise on page 50, is it better to use 7 or 1.17 as our ratio of the lengths when finding the length of? plain our reasoning. 3. In ercise 30 on page 505, a classmate tells ou that our answer is incorrect because ou should have divided the segment into four congruent pieces. Respond to our classmate s argument b justifing our original answer. Performance ask: Pool aintenance Your pool service compan has been ver successful at maintaining rectangular pools designed for regulation competitions. ow ou want to epand our services to owners of private, recreational pools. ow will our eperience transfer to pools of various shapes and sizes? o eplore the answer to this question and more, check out the Performance ask and Real-Life S video at igideasath.com. 507

2 hapter Review namic Solutions available at igideasath.com.1 ilations (pp. 1 ) Graph trapezoid with vertices (1, 1), (1, 3), (3, ), and (3, 1) and its image after a dilation with a scale factor of. Use the coordinate rule for a dilation with k = to find the coordinates of the vertices of the image. hen graph trapezoid and its image. (, ) (, ) (1, 1) (, ) (1, 3) (, ) (3, ) (, ) (3, 1) (, ) Graph the triangle and its image after a dilation with scale factor k. 1. P(, ), Q(, ), R(, ); k = 1. X( 3, ), Y(, 3), Z(1, 1); k = 3 3. You are using a magnifing glass that shows the image of an object that is eight times the object s actual size. he image length is. centimeters. ind the actual length of the object.. Similarit and ransformations (pp. 9 7) escribe a similarit transformation that maps G to L, as shown at the right. G is horizontal, and L is vertical. If ou rotate G 90 about the origin as shown at the bottom right, then the image, G, will have the same orientation as L. L appears to be half as large as G. ilate G using a scale factor of 1. (, ) ( 1, 1 (, ) ( 1, 1) G (, ) G ( 1, 3) (, ) ( 3, ) ) he vertices of G match the vertices of L. So, a similarit transformation that maps G to L is a rotation of 90 about the origin followed b a dilation with a scale factor of 1. L G L G G escribe a similarit transformation that maps to RS.. (1, 0), (, 1), ( 1, ) and R( 3, 0), S(, 3), (3, ) 5. (, ), (, 0), (, ) and R(, 3), S(0, 1), (1, ). (3, ), (0, ), ( 1, 3) and R(, ), S(, 0), (, ) 50 hapter Similarit

3 .3 Similar Polgons (pp. 75 ) In the diagram, G KL. ind the scale factor from G to KL. hen list all pairs of congruent angles and write the ratios of the corresponding side lengths in a statement of proportionalit. rom the diagram, ou can see that and KL K 1 L are corresponding sides. So, the scale factor of 1 G to KL is KL = 1 1 = K, L, G, and. 1 KL = L G = G = K G ind the scale factor. hen list all pairs of congruent angles and write the ratios of the corresponding side lengths in a statement of proportionalit. 7. G. XYZ RPQ G 1 9 X 5 0 Y Z P Q wo similar triangles have a scale factor of 3 : 5. he altitude of the larger triangle is inches. What is the altitude of the smaller triangle? 10. wo similar triangles have a pair of corresponding sides of length 1 meters and meters. he larger triangle has a perimeter of meters and an area of 10 square meters. ind the perimeter and area of the smaller triangle. R. Proving riangle Similarit b (pp. 5 90) etermine whether the triangles are similar. If the are, write a similarit statement. plain our reasoning. ecause the are both right angles, and are congruent. the riangle Sum heorem, m = 10, so m = 9. So, and are congruent. So, b the Similarit heorem. Show that the triangles are similar. Write a similarit statement Q 35 R S U cellular telephone tower casts a shadow that is 7 feet long, while a nearb tree that is 7 feet tall casts a shadow that is feet long. ow tall is the tower? hapter hapter Review 509

4 .5 Proving riangle Similarit b SSS and SS (pp. 93 9) Show that the triangles are similar ompare and b finding ratios of corresponding side lengths. Shortest sides Longest sides Remaining sides = 1 = 7 3 = 35 = 7 3 = 1 = 7 3 ll the ratios are equal, so b the SSS Similarit heorem. Use the SSS Similarit heorem or the SS Similarit heorem to show that the triangles are similar U 9 S 7 R 1 Q. Proportionalit heorems (pp ) In the diagram, bisects. ind the length of. ecause is an angle bisector of, ou can appl the riangle ngle isector heorem. = 5 = = 75 riangle ngle isector heorem Substitute = Solve for. he length of is units. ind the length of ross Products Propert hapter Similarit

5 hapter est etermine whether the triangles are similar. If the are, write a similarit statement. plain our reasoning V 1 U. 110 L 10 3 J 110 K 3. Y X W P Z ind the value of the variable.. 9 w q p 7. Given QRS P, list all pairs of congruent angles. hen write the ratios of the corresponding side lengths in a statement of proportionalit.. here is one slice of a large pizza and one slice of a small pizza in the bo. a. escribe a similarit transformation that maps pizza slice to pizza slice. b. What is one possible scale factor for a medium slice of pizza? plain our reasoning. (Use a dilation on the large slice of pizza.) 9. he original photograph shown is inches b inches. a. What transfomations can ou use to produce the new photograph? b. You dilate the original photograph b a scale factor of 1. What are the dimensions of the new photograph? c. You have a frame that holds photos that are.5 inches b 11 inches. an ou dilate the original photograph to fit the frame? plain our reasoning. new original 10. In a perspective drawing, lines that are parallel in real life must meet at a vanishing point on the horizon. o make the train cars in the drawing appear equal in length, the are drawn so that the lines connecting the opposite corners of each car are parallel. ind the length of the bottom edge of the drawing of ar.. cm vanishing point 19 cm ar 1 5. cm ar hapter hapter est 511

6 umulative ssessment 1. Use the graph of quadrilaterals and QRS. S Q R a. Write a composition of transformations that maps quadrilateral to quadrilateral QRS. b. re the quadrilaterals similar? plain our reasoning.. In the diagram, is a parallelogram. Which congruence theorem(s) could ou use to show that? Select all that appl. SS ongruence heorem SSS ongruence heorem L ongruence heorem S ongruence heorem S ongruence heorem 3. Which epression is not equivalent to ? (5 )( + 7) (5 + )( 7) ( + 7) (5 ). bag contains si tiles with a letter on each tile. he letters are,,,,, and. You randoml draw two tiles from the bag. Which epression represents the probabilit that ou draw the tile and the tile?!! 1 P P 1 51 hapter Similarit

7 5. nter a statement or reason in each blank to complete the two-column proof. Given KJ KL = K K J K Prove L JG G L SS RSOS 1. KJ KL = K K 1. Given. JK LK. 3. JK LK 3.. KJ KL efinition of congruent angles. m KJ + m JG = 10. Linear Pair Postulate 7. m JG = 10 m KJ 7.. m KL + m L = Subtraction Propert of qualit 10. m L = 10 m KJ ransitive Propert of qualit 1. L JG 1.. he coordinates of the vertices of are (, 5), ( 5, ), and ( 1, ). he coordinates of the vertices of JKL are J(1, 10), K(10, 1), and L(, ). J. an ou show that JKL b using the Similarit heorem? If so, do so b listing the congruent corresponding angles and writing a similarit transformation that maps to JKL. If not, eplain wh not. 7. lassif the quadrilateral using the most specific name. rectangle square parallelogram rhombus. Your friend makes the statement Quadrilateral PQRS is similar to quadrilateral WXYZ. escribe the relationships between corresponding angles and between corresponding sides that make this statement true. hapter umulative ssessment 513