Using Ratios and Proportions to Solve Geometry Problems. You can use properties of proportions to solve a variety of algebraic and geometric problems.

Transcription

1 ig Idea T SUY IG IS Using atios and roportions to Solve Geometry roblems You can use properties of proportions to solve a variety of algebraic and geometric problems. or Your otebook 8 or eample, in the diagram above, suppose you know that. Then you can write any of the following relationships. 8 p ig Idea 2 Showing that Triangles are Similar You learned three ways to prove two triangles are similar. Similarity ostulate SSS Similarity Theorem SS Similarity Theorem If > and >, then n, n. If, then n, n. If > and, then n, n. ig Idea 3 Using Indirect easurement and Similarity You can use triangle similarity theorems to apply indirect measurement in order to find lengths that would be inconvenient or impossible to measure directly. onsider the diagram shown. ecause the two triangles formed by the person and the tree are similar by the Similarity ostulate, you can write the following proportion to find the height of the tree. height of person length of person s shadow height of tree length of tree s shadow You also learned about dilations, a type of similarity transformation. In a dilation, a figure is either enlarged or reduced in size. hapter Summary 7

2 T VIW VIW KY VOULY classzone.com ulti-language Glossary Vocabulary practice or a list of postulates and theorems, see pp ratio, p. 3 proportion, p. 38 means, etremes geometric mean, p. 39 scale drawing, p. 3 scale, p. 3 similar polygons, p. 372 scale factor of two similar polygons, p. 373 dilation, p. 09 center of dilation, p. 09 scale factor of a dilation, p. 09 reduction, p. 09 enlargement, p. 09 VOULY XISS opy and complete the statement..? is a transformation in which the original figure and its image are similar. 2. If nq, nxyz, then Q XY? YZ??. 3. WITIG escribe the relationship between a ratio and a proportion. Give an eample of each. VIW XLS XISS Use the review eamples and eercises below to check your understanding of the concepts you have learned in each lesson of hapter.. atios, roportions, and the Geometric ean pp X L The measures of the angles in n are in the etended ratio of 3 : :. ind the measures of the angles. Use the etended ratio of 3 : : to label the angle measures as 38, 8, and Triangle Sum Theorem ombine like terms. ivide each side by 2. So, the angle measures are 3(8) 8, (8) 08, and (8) 78. XLS, 3, and on pp for s. XISS. The length of a rectangle is meters and the width is meters. ind the ratio of the width to the length of the rectangle. Then simplify the ratio.. The measures of the angles in nuvw are in the etended ratio of : : 2. ind the measures of the angles.. ind the geometric mean of 8 and 2. 8 hapter Similarity

3 classzone.com hapter eview ractice.2 Use roportions to Solve Geometry roblems pp X L In the diagram,. ind Substitution roperty of quality Solve for. ross roducts roperty 3 2 XL 2 on p. 3 for s. 7 8 XISS Use the diagram and the given information to find the unknown length. 7. Given Q QL, find. 8. Given, find. L Use Similar olygons pp X L In the diagram, G, KL. ind the scale factor. rom the diagram, you can see that and KL correspond. So, the scale factor of G to KL is 2 2. KL K G 2 8 L 2 XLS 2 and on pp for s. 9 XISS In ercises 9 and 0, determine whether the polygons are similar. If they are, write a similarity statement and find the scale factor G 0. 9 X 2 0. OSTS Two similar posters have a scale factor of :. The large poster s perimeter is 8 inches. ind the small poster s perimeter. Y Z 8 hapter eview 9

4 rove. T VIW Triangles Similar by pp X L etermine whether the triangles are similar. If they are, write a similarity statement. plain your reasoning. ecause they are right angles, >. y the Triangle Sum Theorem, m 808, so m 298 and >. Then, two angles of n are congruent to two angles of n. So, n, n XLS 2 and 3 on pp for s. 2 XISS Use the Similarity ostulate to show that the triangles are similar S U 38 T LL TOW cellular telephone tower casts a shadow that is 72 feet long, while a tree nearby that is 27 feet tall casts a shadow that is feet long. ow tall is the tower?. rove Triangles Similar by SSS and SS pp X L Show that the triangles are similar. otice that the lengths of two pairs of corresponding sides are proportional. WZ 2 VZ 2 YZ 2 3 XZ 30 3 X Y 30 2 Z W V The included angles for these sides, XZY and VZW, are vertical angles, so XZY > VZW. Then nxyz, nvwz by the SS Similarity Theorem. XL on p. 39 for s. XISS Use the SSS Similarity Theorem or SS Similarity Theorem to show that the triangles are similar T. U 9 S 7 hapter Similarity

5 classzone.com hapter eview ractice. Use roportionality Theorems pp X L etermine whether i LQ. egin by finding and simplifying ratios of lengths determined by L Q 2 L ecause L Q, is parallel to LQ by Theorem., the Triangle roportionality onverse. XL 2 on p. 398 for s. 7 8 XISS Use the given information to determine whether i erform Similarity Transformations pp. 09 X L raw a dilation of quadrilateral GJ with vertices (, ), G(2, 2), (, ), and J(2, 2). Use a scale factor of 2. irst draw GJ. ind the dilation of each verte by multiplying its coordinates by 2. Then draw the dilation. (, y) (2, 2y) (, ) (2, 2) G(2, 2) (, ) (, ) (8, 2) J(2, 2) (, 22) y G J XLS and 2 on pp for s. 9 2 XISS raw a dilation of the polygon with the given vertices using the given scale factor k. 9. T(0, 8), U(, 0), V(0, 0); k 3 2. (, 0), (3, 9), (0, 0), (3, ); k 2. (8, 2), Q(, 0), (3, ), S(, ); k 0. hapter eview 2

6 Solve T TST the proportion y b a 8 a 2 In ercises 7, use the diagram where nq, n.. List all pairs of congruent angles.. Write the ratios of the corresponding sides in a statement of proportionality. 7. ind the value of. etermine whether the triangles are similar. If so, write a similarity statement and the postulate or theorem that justifies your answer L X 8 Y Z K 27 L 9 8 J In ercises 3, find the length of etermine whether the dilation from igure to igure is a reduction or an enlargement. Then find its scale factor.. y. y 9. SL OL You are making a scale model of your school s baseball diamond as part of an art project. The distance between two consecutive bases is 90 feet. If you use a scale factor of to build your 80 model, what will be the distance around the bases on your model? 22 hapter Similarity

7 SOLV QUTI QUTIOS SILIY ILS LG VIW classzone.com lgebra radical epression is simplified when the radicand has no perfect square factor ecept, there is no fraction in the radicand, and there is no radical in a denominator. X L Solve quadratic equations by finding square roots Solve the equation Write original equation. dd 3 to each side ivide each side by. 28 ab ap b, so 28 p Simplify. X L 2 Simplify quotients with radicals Simplify the epression. a. Î 0 8 b. Î Solution a. Î 0 8 Î Simplify fraction. Î a b a b. 2 Simplify. b. Î Î a b a b.. p ultiply numerator and denominator by. ultiply fractions. ap a a. XISS XL for s. 9 Solve the equation or write no solution ( 2 2 7) (2 2 ) 39 XL 2 for s. 0 7 Simplify the epression. 0. Î 7 8. Î 3 2. Î Î Î 2 2 lgebra eview 23

8 Standardized TST TIO Scoring ubric ull redit solution is complete and correct artial redit solution is complete but has errors, or solution is without error but is incomplete o redit no solution is given, or solution makes no sense XT SOS QUSTIOS O L To find the height of a tree, a student 3 inches in height measures the length of the tree s shadow and the length of his own shadow, as shown. The student casts a shadow 8 inches in length and the tree casts a shadow 77 inches in length. a. plain why nq, ntqs. b. ind the height of the tree. c. Suppose the sun is a little lower in the sky. an you still use this method to measure the height of the tree? plain. elow are sample solutions to the problem. ead each solution and the comments in blue to see why the sample represents full credit, partial credit, or no credit. SL : ull credit solution The reasoning is complete. The proportion and calculations are correct. a. ecause they are both right angles, Q > QTS. lso, Q > Q by the efleive roperty. So, nq, ntqs by the Similarity ostulate. b. TS Q TQ 3 TS (77) 8pTS 37 TS In part (b), the question is answered correctly. In part (c), the reasoning is complete and correct. The height of the tree is 37 inches. c. s long as the sun creates two shadows, I can use this method. ngles and T will always be right angles. The measure of Q will change as the sun s position changes, but the angle will still be congruent to itself. So, nq and ntqs will still be similar, and I can write a proportion. 2 hapter Similarity

9 SL 2: artial credit solution In part (a), there is no eplanation of why the postulate can be applied. In part (b), the proportion is incorrect, which leads to an incorrect solution. In part (c), a partial eplanation is given. a. nq, ntqs by the ngle-ngle Similarity ostulate. b. TS Q T 3 TS TS The height of the tree is 308 inches. c. s long as the sun creates two shadows, I can use this method because the triangles will always be similar. SL 3: o credit solution The reasoning in part (a) is incomplete. In part (b), no work is shown. The answer in part (c) is incorrect. a. The triangles are similar because the lines are parallel and the angles are congruent. b. TS 37 inches c. o. The angles in the triangle will change, so you can t write a proportion. TI pply the Scoring ubric. student s solution to the problem on the previous page is given below. Score the solution as full credit, partial credit, or no credit. plain your reasoning. If you choose partial credit or no credit, eplain how you would change the solution so that it earns a score of full credit. a. Q > TS, and Q is in both triangles. So, nq, ntqs. b. QT Q ST (77) ø 3.3 The tree is about 3.3 inches tall. c. The method will still work because the triangles will still be similar if the sun changes position. The right angles will stay right angles, and Q is in both triangles, so it does not matter if its measure changes. Standardized Test reparation 2

10 Standardized TST TI XT SOS. Use the diagram. a. plain how you know that n, n. b. ind the value of n. c. The perimeter of n is 22. What is the perimeter of n? Justify your answer. 0 n 2. On the easel shown at the right, i i G, and G >. a. ind,, and. Justify your answer. b. On the easel, is a support bar attached to,, and G. On this support bar, 0 inches. ind the length of to the nearest inch. Justify your answer. c. The support bar bisects,, and G. oes this mean that polygons and G are similar? plain. in. G 30 in. 3. handmade rectangular rug is available in two sizes at a rug store. small rug is 2 inches long and inches wide. large rug is 3 inches long and 2 inches wide. a. re the rugs similar? If so, what is the ratio of their corresponding sides? plain. b. ind the perimeter and area of each rug. Then find the ratio of the perimeters (large rug to small rug) and the ratio of the areas (large rug to small rug). c. It takes feet of wool yarn to make square foot of either rug. ow many inches of yarn are used for each rug? plain. d. The price of a large rug is. times the price of a small rug. The store owner wants to change the prices for the rugs, so that the price for each rug is based on the amount of yarn used to make the rug. If the owner changes the prices, about how many times as much will the price of a large rug be than the price of a small rug? plain.. In the diagram shown at the right, ] OQ passes through the origin. a. plain how you know that nos, noq. b. ind the coordinates of point Q. Justify your answer. c. The -coordinate of a point on ] OQ is a. Write the y-coordinate of this point in terms of a. Justify your answer. y (, 3) O S(, 0) Π(9, 0) 2 hapter Similarity

11 STT TST TI classzone.com ULTIL OI. If nq, nstu, which proportion is not necessarily true? Q Q ST TU Q SU TU SU Q TU Q ST SU GI SW 8. ind the value of. J K L 9. On a map, the distance between two cities is 2 3 inches. The scale on the map is in.:80 mi. What is the actual distance between the two cities? 0 mi 0 mi 80 mi 2 mi 7. In the diagram, what is the scale factor of the dilation from nq to ntuv? 9. In the diagram below, nq, n, and > Q. If 2, find. 0. Given G 0, find. Π3 y 2 U G T 2 V 2 3. In an acute isosceles triangle, the measures of two of the angles are in the ratio :. ind the measure of a base angle in the triangle. 2 3 SOT SOS 2. On a school campus, the gym is 00 feet from the art studio. a. Suppose you draw a map of the school campus using a scale of inch: 00 feet. ow far will the gym be from the art studio on your map? b. Suppose you draw a map of the school campus using a scale of 2 inch : 00 feet. Will the distance from the gym to the art studio on this map be greater than or less than the distance on the map in part (a)? plain. 3. ectangles and G are similar, and the ratio of to is : 3. In each rectangle, the length is twice the width. The area of is 32 square inches. ind the length, width, and area of G. plain. Standardized Test ractice 27

12 UULTIV VIW hapters ind m 2 if and 2 are (a) complementary angles and (b) supplementary angles. (p. 2). m m m 888. m 8 Solve the equation and write a reason for each step. (p. 0) ( 2 3) ( 2) 2 State the postulate or theorem that justifies the statement. (pp. 7, ) 8. > > 0. m 3 m > > 3 3. m 7 m p 2 l m The variable epressions represent the angle measures of a triangle. ind the measure of each angle. Then classify the triangle by its angles. (p. 27). m 8. m 28. m (3 2 )8 m 38 m 28 m ( )8 m 8 m ( 2 )8 m ( 2 )8 etermine whether the triangles are congruent. If so, write a congruence statement and state the postulate or theorem you used. (pp. 23, 20, 29) V 9. Y W X S Z ind the value of. (pp. 29, 303, 30). 2. Y X W 2 9 Z ( 2)8 (3 2 28)8 etermine whether the triangles are similar. If they are, write a similarity statement and state the postulate or theorem you used. (pp. 38, 388) G Y Z V W X 2 28 G 3 28 S 28 umulative eview: hapters

13 2. OITS company s profits for two years are shown in the table. lot and connect the points (, y). Use the idpoint ormula to estimate the company s profits in 03. (ssume that profits followed a linear pattern.) (p. ) Years since 00, rofit, y (in dollars) 2,000 3, 27. TIS SI The graph at the right models the accumulated cost for an individual adult tennis club membership for several months. (p. 80) a. Write an equation of the line. b. Tell what the slope and y-intercept mean in this situation. c. ind the accumulated cost for one year. ost (dollars) y 70 (9, 78) 00 (0, ) onth OO Write a two-column proof or a paragraph proof. (pp. 23, 20, 29) 28. GIV c G > J, > KG, 29. GIV c i J, KJ J > OV c n > njgk OV c n > n G J L K 30. OUITY T building committee needs to choose a site for a new community center. The committee decides that the new center should be located so that it is the same distance from each of the three local schools. Use the diagram to make a sketch of the triangle formed by the three schools. plain how you can use this triangle to locate the site for the new community center. (p. 303) ranklin Washington onroe 3. GOGY The map shows the distances between three cities in orth akota. escribe the range of possible distances from owman to llendale. (p. 328) 32. L You send 2 photos to a company that makes personalized wall calendars. The company enlarges the photos and inserts one for each month on the calendar. ach photo is inches by inches. The image for each photo on the calendar is 0 inches by inches. What is the scale factor of the enlargement? (p. 09) umulative eview: hapters 29