Identify similar figures. Solve problems involving scale factors. do artists use geometric patterns?

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1 imilar Polygons Identify similar figures. olve problems involving scale factors. Vocabulary similar polygons scale factor do artists use geometric patterns? M.. scher (19 19) was a utch graphic artist known for drawing impossible structures, spatial illusions, and repeating interlocking geometric patterns. The image at the right is a print of scher s ircle Limit IV, which is actually a woodcutting. It includes winged images that have the same shape, but are different in size. lso note that there are not only similar dark images but also similar light images. ircle Limit IV, M.. scher (190) 00 ordon rt.v., aarn, olland. ll rights reserved. INTIY IMIL IU When polygons have the same shape but may be different in size, they are called. similar polygons tudy Tip imilarity and ongruence If two polygons are congruent, they are also similar. ll of the corresponding angles are congruent, and the lengths of the corresponding sides have a ratio of 1:1. Words imilar Polygons Two polygons are similar if and only if their corresponding angles are congruent and the measures of their corresponding sides are proportional. ymbol is read is similar to xample m m m m m. m m m The order of the vertices in a similarity statement is important. It identifies the corresponding angles and the corresponding sides. similarity statement congruent angles corresponding sides Like congruent polygons, similar polygons may be repositioned so that corresponding parts are easy to identify. Lesson - imilar Polygons 9 rt esource, NY

2 tudy Tip Identifying orresponding Parts Using different colors to circle letters of congruent angles may help you identify corresponding parts. xample 1 imilar Polygons etermine whether each pair of figures is similar. Justify your answer. a ll right angles are congruent, so. ince m m,. y the Third ngle Theorem,. Thus, all corresponding angles are congruent. Now determine whether corresponding sides are proportional. ides opposite 90 angle ides opposite 0 angle ides opposite 0 angle 1 or or 1. or 1.. The ratios of the measures of the corresponding sides are equal, and the corresponding angles are congruent, so. tudy Tip ommon Misconception When two figures have vertices that are in alphabetical order, this does not mean that the corresponding vertices in the similarity statement will follow alphabetical order. b. oth rectangles have all right angles and right angles are congruent. and, but because. The rectangles are not similar. L TO When you compare the lengths of corresponding sides of similar figures, you usually get a numerical ratio. This ratio is called the scale factor for the two figures. cale factors are often given for models of real-life objects. xample cale actor MOVI ome special effects in movies are created using miniature models. In a recent movie, a model sports-utility vehicle (UV) inches long was created to look like a real 1 -foot UV. What is the scale factor of the model compared to the real UV? efore finding the scale factor you must make sure that both measurements use the same unit of measure. length of model length of real UV 1 (1) 1 inches inches 1 inches 1 The ratio comparing the two lengths is 1 that the model is 1 the length of the real UV. or 1:. The scale factor is 1, which means 90 hapter Proportions and imilarity

3 When finding the scale factor for two similar polygons, the scale factor will depend on the order of comparison. cm cm 1 cm cm cm cm cm. cm The scale factor of quadrilateral to quadrilateral is. The scale factor of quadrilateral to quadrilateral is 1. xample Proportional Parts and cale actor The two polygons are similar. a. Write a similarity statement. Then find x, y, and UT. Use the congruent angles to write the corresponding vertices in order. polygon TUV polygon Now write proportions to find x and y. To find x: To find y: T V T UT imilarity proportion 1 x T 1, 1 y V x, 1 m T (y ) m x m V U m imilarity proportion T 1, UT y, m m 1() (x) ross products 1() (y ) ross products x Multiply. 90 y Multiply. 1. x ivide each side by. y ubtract from each side. 0. y ivide each side by. UT y, so UT 0. or.. tudy Tip hecking olutions To verify the scale factor, find the ratio of two other corresponding sides. b. ind the scale factor of polygon TUV to polygon. The scale factor is the ratio of the lengths of any two corresponding sides. T 1 9 or You can use scale factors to produce similar figures. xample nlargement of a igure Triangle is similar to XYZ with a scale factor of. If the lengths of the sides of are,, and inches, what are the lengths of the sides of XYZ? Write proportions for finding side measures. XYZ x y XYZ 1 0 XYZ z 1 x y 0 z 9 x 1 y 1 z The lengths of the sides of XYZ are 9, 1, and 1 inches. Lesson - imilar Polygons 91

4 xample cale actors on Maps MP The scale on the map of New Mexico is centimeters miles. The distance on the map across New Mexico from east to west through lbuquerque is.1 centimeters. ow long would it take to drive across New Mexico if you drove at an average of 0 miles per hour? lbuquerque oswell xplore very centimeters represents miles. The distance across the map is.1 centimeters. Las ruces Plan reate a proportion relating the measurements to the scale to find the distance in miles. Then use the formula d rt to find the time. olve centimeters miles x centimeters miles x ross products tudy Tip Units of Time emember that there are 0 minutes in an hour. When rewriting as a 0 mixed number, you could also write, which 0 means hours minutes. x ivide each side by. The distance across New Mexico is approximately miles. d rt 0t d and r 0 t 0 ivide each side by 0. t 1 implify. It would take hours or about hours and minutes to drive 1 across New Mexico at an average of 0 miles per hour. xamine eexamine the scale. If centimeters miles, then centimeters 0 miles. The map is about centimeters wide, so the distance across New Mexico is about 0 miles. The answer is about. hours and at 0 miles per hour, the trip would be 0 miles. The two distances are close estimates, so the answer is reasonable. oncept heck 1. IN T O oberto and arrett have calculated their scale factor for two similar triangles. oberto PQ 1 0 arrett PQ 1 0 in. P in. Q 9 hapter Proportions and imilarity Who is correct? xplain your reasoning.

5 . ind a counterexample for the statement ll rectangles are similar.. OPN N xplain whether two polygons that are congruent are also similar. Then explain whether two polygons that are similar are also congruent. uided Practice etermine whether each pair of figures is similar. Justify your answer... P 0 Q I ach pair of polygons is similar. Write a similarity statement, and find x, the measure(s) of the indicated side(s), and the scale factor...,, and 1 x 1. rectangle with length 0 centimeters and height 0 centimeters is reduced so that the new rectangle is similar to the original and the scale factor is 1. ind the length and height of the new rectangle. 9. triangle has side lengths of meters, meters, and meters. The triangle is enlarged so that the larger triangle is similar to the original and the scale factor is. ind the perimeter of the larger triangle. 1 1 x x pplication. MP efer to xample on page 9. raw the state of New Mexico using a scale of centimeters 0 miles. Is your drawing similar to the one in xample? xplain how you know. Practice and pply or xercises , 9 1 ee xamples 1, xtra Practice ee page. etermine whether each pair of figures is similar. Justify your answer X 0 0 Z W M P.1.9. Y N Lesson - imilar Polygons 9

6 1. ITTU The replica of the iffel Tower at an amusement park is 0 feet tall. The actual iffel Tower is feet tall. What is the scale factor comparing the amusement park tower to the actual tower? 1. POTOOPYIN Mr. ichardson walked to a copier in his office, made a copy of his proposal, and sent the original to one of his customers. When Mr. ichardson looked at his copy before filing it, he saw that the copy had been made at an 0% reduction. e needs his filing copy to be the same size as the original. What enlargement scale factor must he use on the first copy to make a second copy the same size as the original? ach pair of polygons is similar. Write a similarity statement, and find x, the measures of the indicated sides, and the scale factor. 1. and 1. and x + 1 x 1 x and 0. and 1 x x +. x T 1 x POTOPY or xercises 1, use the following information. picture is enlarged by a scale factor of and then enlarged again by the same factor. 1. If the original picture was. inches by inches, what were its dimensions after both enlargements?. Write an equation describing the enlargement process.. y what scale factor was the original picture enlarged? POT Make a scale drawing of each playing field using the given scale.. Use the information about the soccer field in rew tadium. Use the scale 1 millimeter 1 meter.. basketball court is feet by 0 feet. Use the scale 1 inch feet.. tennis court is feet by feet. Use the scale 1 inch 1 foot. ports rew tadium in olumbus, Ohio, was specifically built for Major League occer. The dimensions of the field are about 9 meters by meters. ource: etermine whether each statement is always, sometimes, or never true.. Two congruent triangles are similar.. Two squares are similar. 9. triangle is similar to a quadrilateral. 0. Two isosceles triangles are similar. 1. Two rectangles are similar.. Two obtuse triangles are similar.. Two equilateral triangles are similar. 9 hapter Proportions and imilarity Joe iblin/olumbus rew/ml

7 ach pair of polygons is similar. ind x and y. ound to the nearest hundredth if necessary.. I. K M N 9 1 y 0 (y 0) x 0 Q J L O 0 (x ) J L.. x 1 y 1 x y x 0 y 1 1 y x T 9 W 0 9 V 1 U or xercises 0, use the following information to find each measure. Polygon polygon, m, 1, 1,.,, and. 0. scale factor of trapezoid to trapezoid 1... m.. perimeter of trapezoid. perimeter of trapezoid. ratio of the perimeter of polygon to the perimeter of polygon etermine which of the following right triangles are similar. Justify your answer. J 0. L 1. 1 K 0 9 N M. 1 O 1. P 1 0. I OOINT OMTY raph the given points. raw polygon and M N. ind the coordinates for vertices L and P such that NLPM. 9. (, 0), (, ), (0, ), (, 0); M(, 0), N(1, 0) 0. (, 1), (, ), (, 0), (, ); M(, 1), N 1 1, Lesson - imilar Polygons 9

8 ONTUTION or xercises 1 and, use the following information. floor plan is given for the first floor of a new house. One inch represents feet. Use the information in the plan to find the dimensions. 1. living room. deck in. in. 1 1 in. Living eck in. oom Kitchen Master uite ining oom ITIL TINKIN or xercises, use the following information. The area of a rectangle is the product of its length and width w. ectangle is similar to rectangle WXYZ with sides in a ratio of :1.. What is the ratio of the areas of the two rectangles?. uppose the dimension of each rectangle is tripled. What is the new ratio of the sides of the rectangles?. What is the ratio of the areas of these larger rectangles? TTITI or xercises, refer to the graphic, which uses rectangles to represent percents.. re the rectangles representing % and 1% similar? xplain.. What is the ratio of the areas of the rectangles representing % and 1% if area length width? ompare the ratio of the areas to the ratio of the percents. U TOY napshots Workplace manners declining survey asked workers whether they thought the level of professional courtesy in the workplace had increased or decreased in the past five years. Their responses: ecreased %. Use the graph to make a conjecture about the overall changes in the level of professional courtesy in the workplace in the past five years. Increased Not changed on t know/no answer ource: OfficeTeam poll of adults eb. -1. Margin of error: +/. percentage points. % 1% % y am Ward, U TOY ITIL TINKIN or xercises 9 and 0,. 9. how that the perimeters of and have the same ratio as their corresponding sides. 0. If units are added to the lengths of each side, are the new triangles similar? b a c b a c 9 hapter Proportions and imilarity 1. WITIN IN MT nswer the question that was posed at the beginning of the lesson. ow do artists use geometric patterns? Include the following in your answer: why scher called the picture ircle Limit IV, and how one of the light objects and one of the dark objects compare in size.

9 tandardized Test Practice. In a history class with students, the ratio of girls to boys is to. ow many more girls are there than boys? 1 1. L ind x m 1.1 m m 9. m x m 1 m 1 xtending the Lesson cale factors can be used to produce similar figures. The resulting figure is an enlargement or reduction of the original figure depending on the scale factor. Triangle has vertices (0, 0), (, 0), and (, ). uppose the coordinates of each vertex are multiplied by to create the similar triangle.. ind the coordinates of the vertices of.. raph and.. Use the istance ormula to find the measures of the sides of each triangle.. ind the ratios of the sides that appear to correspond.. ow could you use slope to determine if angles are congruent? 9. Is? xplain your reasoning. Maintain Your kills Mixed eview olve each proportion. (Lesson -1) b c c. y y Use the figure to write an inequality relating each pair of angle or segment measures. (Lesson -). O, O. m O, m O. m, m 0 O 9.. ind x. (Lesson -)... x 0 x x etting eady for the Next Lesson 9. uppose two parallel lines are cut by a transversal and 1 and are alternate interior angles. ind m 1 and m if m 1 x 9 and m 9x. (Lesson -) PQUIIT KILL In the figure,,, and m 11. ind the measure of each angle. (To review angles and parallel lines, see Lesson -.) Lesson - imilar Polygons 9