Similar Polygons. Essential Question How are similar polygons related? Work with a partner. Use dynamic geometry software to draw any ABC.

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1 .1 imilar olygons OO O earning tandard HG-T..2 HG-G.. ssential uestion How are similar polygons related? omparing Triangles after a ilation Work with a partner. Use dynamic geometry software to draw any. ilate to form a similar using any scale factor k and any center of dilation. a. ompare the corresponding angles of and. b. ind the ratios of the lengths of the sides of to the lengths of the corresponding sides of. What do you observe? c. epeat parts (a) and (b) for several other triangles, scale factors, and centers of dilation. o you obtain similar results? omparing Triangles after a ilation OOIG O TUTU Work with a partner. Use dynamic geometry software to draw any. ilate to form a similar using any scale factor k and any center of dilation. To be proficient in math, you need to look closely to discern a pattern or structure. a. ompare the perimeters of and. What do you observe? b. ompare the areas of and. What do you observe? c. epeat parts (a) and (b) for several other triangles, scale factors, and centers of dilation. o you obtain similar results? ommunicate Your nswer. How are similar polygons related?. T is dilated by a scale factor of to form T. The area of T is 1 square inch. What is the area of T? ection.1 imilar olygons 17

2 .1 esson What You Will earn ore Vocabulary revious similar figures similarity transformation corresponding parts OOIG O TUTU otice that any two congruent figures are also similar. In and WXY below, the scale factor is = = 7 = 1. o, you can 7 write WXY and WXY. 7 W X 7 Y Use similarity statements. ind corresponding lengths in similar polygons. ind perimeters and areas of similar polygons. ecide whether polygons are similar. Using imilarity tatements ecall from ection. that two geometric figures are similar figures if and only if there is a similarity transformation that maps one figure onto the other. ore oncept orresponding arts of imilar olygons In the diagram below, is similar to. You can write is similar to as. similarity transformation preserves angle measure. o, corresponding angles are congruent. similarity transformation also enlarges or reduces side lengths by a scale factor k. o, corresponding side lengths are proportional. a b orresponding angles,, c similarity transformation ka kb kc atios of corresponding side lengths = = = k IG In a statement of proportionality, any pair of ratios forms a true proportion. Using imilarity tatements In the diagram, T XYZ. a. ind the scale factor from T to XYZ. b. ist all pairs of congruent angles. c. Write the ratios of the corresponding side lengths in a statement of proportionality. OUTIO a. XY = = YZ T = 1 0 = T ZX T = 1 2 = X Z Y o, the scale factor is. b. X, Y, and T Z. c. ecause the ratios in part (a) are equal, XY = YZ T = ZX T. onitoring rogress Help in nglish and panish at igideasath.com 1. In the diagram,. ind the scale factor from to. Then list all pairs of congruent angles and write the ratios of the corresponding side lengths in a statement of proportionality. 1 hapter imilarity

3 IG orresponding lengths in similar triangles include side lengths, altitudes, medians, and midsegments. inding orresponding engths in imilar olygons ore oncept orresponding engths in imilar olygons If two polygons are similar, then the ratio of any two corresponding lengths in the polygons is equal to the scale factor of the similar polygons. inding a orresponding ength In the diagram,. ind the value of. IIG TY OIT There are several ways to write the proportion. or eample, you could write =. OUTIO The triangles are similar, so the corresponding side lengths are proportional. = Write proportion. 1 1 = 0 ubstitute. 1 = 0 ross roducts roperty = 2 olve for. The value of is inding a orresponding ength In the diagram, T XZ. ind the length of the altitude. OUTIO irst, find the scale factor from T to XZ. T XZ = + + = 1 = ecause the ratio of the lengths of the altitudes in similar triangles is equal to the scale factor, you can write the following proportion. Y = Write proportion. = ubstitute for Y. = 1 ultiply each side by and simplify. The length of the altitude is 1. T X Y Z onitoring rogress 2. ind the value of.. ind. Help in nglish and panish at igideasath.com 10 1 T G 0 H T G ection.1 imilar olygons 1

4 YZIG TIOHI When two similar polygons have a scale factor of k, the ratio of their perimeters is equal to k. inding erimeters and reas of imilar olygons Theorem Theorem.1 erimeters of imilar olygons If two polygons are similar, then the ratio of their perimeters is equal to the ratios of their corresponding side lengths If, then = = = =. roof. 2, p. 2; igideasath.com odeling with athematics town plans to build a new swimming pool. n Olympic pool is rectangular with a length of 0 meters and a width of 2 meters. The new pool will be similar in shape to an Olympic pool but will have a length of 0 meters. ind the perimeters of an Olympic pool and the new pool. 2 m 0 m TUY TI You can also write the scale factor as a decimal. In ample, you can write the scale factor as 0. and multiply by 10 to get = 0.(10) = 0. OUTIO 1. Understand the roblem You are given the length and width of a rectangle and the length of a similar rectangle. You need to find the perimeters of both rectangles. 2. ake a lan ind the scale factor of the similar rectangles and find the perimeter of an Olympic pool. Then use the erimeters of imilar olygons Theorem to write and solve a proportion to find the perimeter of the new pool.. olve the roblem ecause the new pool will be similar to an Olympic pool, the scale factor is the ratio of the lengths, 0 0 =. The perimeter of an Olympic pool is 2(0) + 2(2) = 10 meters. Write and solve a proportion to find the perimeter of the new pool. 10 = = 0 erimeters of imilar olygons Theorem ultiply each side by 10 and simplify. o, the perimeter of an Olympic pool is 10 meters, and the perimeter of the new pool is 0 meters. Gazebo 10 m Gazebo 1 m 1 m 1 m G m H m. ook ack heck that the ratio of the perimeters is equal to the scale factor = onitoring rogress Help in nglish and panish at igideasath.com. The two gazebos shown are similar pentagons. ind the perimeter of Gazebo. hapter imilarity

5 YZIG TIOHI When two similar polygons have a scale factor of k, the ratio of their areas is equal to k 2. Theorem Theorem.2 reas of imilar olygons If two polygons are similar, then the ratio of their areas is equal to the squares of the ratios of their corresponding side lengths. rea of If, then rea of ( = roof., p. 2; igideasath.com ) 2 = ( ) 2 = ( ) 2 = ( ) 2. inding reas of imilar olygons In the diagram,. ind the area of. 10 cm cm OUTIO rea of = cm 2 ecause the triangles are similar, the ratio of the area of to the area of is equal to the square of the ratio of to. Write and solve a proportion to find the area of. et represent the area of. rea of rea of ( = ) 2 ( = 10 2 ) = reas of imilar olygons Theorem ubstitute. quare the right side of the equation. 2 = 100 ross roducts roperty 00 = 100 implify. = olve for. The area of is square centimeters. onitoring rogress. In the diagram, GH. ind the area of. Help in nglish and panish at igideasath.com G 7 m 21 m H rea of GH = m 2 ection.1 imilar olygons 21

6 eciding Whether olygons re imilar eciding Whether olygons re imilar ecide whether and are similar. plain your reasoning. OUTIO orresponding sides of the pentagons are proportional with a scale factor of 2. However, this does not necessarily mean the pentagons are similar. dilation with center and scale factor 2 moves onto GH. Then a reflection moves GH onto. G H does not eactly coincide with, because not all the corresponding angles are congruent. (Only and are congruent.) ecause angle measure is not preserved, the two pentagons are not similar. onitoring rogress Help in nglish and panish at igideasath.com efer to the floor tile designs below. In each design, the red shape is a regular heagon. Tile esign 1 Tile esign 2. ecide whether the heagons in Tile esign 1 are similar. plain. 7. ecide whether the heagons in Tile esign 2 are similar. plain. 22 hapter imilarity

7 .1 ercises ynamic olutions available at igideasath.com Vocabulary and ore oncept heck 1. OT TH T or two figures to be similar, the corresponding angles must be, and the corresponding side lengths must be. 2. IT WO, UTIO Which is different? ind both answers. What is the scale factor? 1 What is the ratio of their areas? What is the ratio of their corresponding side lengths? What is the ratio of their perimeters? onitoring rogress and odeling with athematics In ercises and, find the scale factor. Then list all pairs of congruent angles and write the ratios of the corresponding side lengths in a statement of proportionality. (ee ample 1.)...7. G G 1 2 In ercises, the polygons are similar. ind the value of. (ee ample 2.) H 1 1 G G H ection.1 imilar olygons 2

8 In ercises and 10, the black triangles are similar. Identify the type of segment shown in blue and find the value of the variable. (ee ample.) OIG WITH THTI Your family has decided to put a rectangular patio in your backyard, similar to the shape of your backyard. Your backyard has a length of feet and a width of feet. The length of your new patio is 1 feet. ind the perimeters of your backyard and of the patio. In ercises 1 22, the polygons are similar. The area of one polygon is given. ind the area of the other polygon. (ee ample.) 10. y ft = 27 ft 2 ft y 1 In ercises 11 and, TU. ind the ratio of their perimeters U 1 T cm = 10 cm 2 in. cm in. = 100 in. 2 U T cm cm = cm 2 In ercises 1 1, two polygons are similar. The perimeter of one polygon and the ratio of the corresponding side lengths are given. ind the perimeter of the other polygon. 1. perimeter of smaller polygon: cm; ratio: 2 1. perimeter of smaller polygon: ft; ratio: 1. perimeter of larger polygon: 0 yd; ratio: 1 1. perimeter of larger polygon: m; ratio: OIG WITH THTI school gymnasium is being remodeled. The basketball court will be similar to an basketball court, which has a length of feet and a width of 0 feet. The school plans to make the width of the new court feet. ind the perimeters of an court and of the new court in the school. (ee ample.) 2. O YI escribe and correct the error in finding the perimeter of triangle. The triangles are similar = 2 = = 2. O YI escribe and correct the error in finding the area of rectangle. The rectangles are similar. = 2 units = 2 = 2 = 72 2 hapter imilarity

9 In ercises 2 and 2, decide whether the red and blue polygons are similar. (ee ample.) WIG OUIO In table tennis, the table is a rectangle feet long and feet wide. tennis court is a rectangle 7 feet long and feet wide. re the two surfaces similar? plain. If so, find the scale factor of the tennis court to the table THTI OTIO In ercises 7 and, the two polygons are similar. ind the values of and y. 27. OIG Triangles and are similar. Which statement is correct? elect all that apply. = = = = y 1 YZIG TIOHI In ercises 2, GH. H G y z ind the scale factor of to GH. 2. ind the scale factor of GH to. 0. ind the values of, y, and z. 1. ind the perimeter of each polygon. 2. ind the ratio of the perimeters of to GH.. ind the area of each polygon.. ind the ratio of the areas of to GH.. UIG TUTU ectangle is similar to rectangle. ectangle has side lengths of and. ectangle has a side length of 1. What are the possible values for the length of the other side of rectangle? elect all that apply. 2. (y 7) TTIG TO IIO In ercises 2, the figures are similar. ind the missing corresponding side length.. igure has a perimeter of 72 meters and one of the side lengths is 1 meters. igure has a perimeter of 0 meters. 0. igure has a perimeter of 2 inches. igure has a perimeter of inches and one of the side lengths is inches. 1. igure has an area of square feet and one of the side lengths is feet. igure has an area of 7 square feet. 2. igure has an area of 1 square feet. igure has an area of square feet and one of the side lengths is 1 feet. ection.1 imilar olygons 2

10 ITI THIIG In ercises, tell whether the polygons are always, sometimes, or never similar.. two isosceles triangles. two isosceles trapezoids. two rhombuses. two squares 7. two regular polygons. a right triangle and an equilateral triangle. IG GUT Your sister claims that when the side lengths of two rectangles are proportional, the two rectangles must be similar. Is she correct? plain your reasoning. 0. HOW O YOU IT? You shine a flashlight directly on an object to project its image onto a parallel screen. Will the object and the image be similar? plain your reasoning. 2. OVIG THO rove the erimeters of imilar olygons Theorem (Theorem.1) for similar rectangles. Include a diagram in your proof.. OVIG THO rove the reas of imilar olygons Theorem (Theorem.2) for similar rectangles. Include a diagram in your proof.. THOUGHT OVOIG The postulates and theorems in this book represent uclidean geometry. In spherical geometry, all points are points on the surface of a sphere. line is a circle on the sphere whose diameter is equal to the diameter of the sphere. plane is the surface of the sphere. In spherical geometry, is it possible that two triangles are similar but not congruent? plain your reasoning.. ITI THIIG In the diagram, is a square, and. ind the eact value of. This value is called the golden ratio. Golden rectangles have their length and width in this ratio. how that the similar rectangles in the diagram are golden rectangles OIG WITH THTI uring a total eclipse of the un, the moon is directly in line with the un and blocks the un s rays. The distance between arth and the un is,000,000 miles, the distance between arth and the moon is,000 miles, and the radius of the un is 2,00 miles. Use the diagram and the given measurements to estimate the radius of the moon. un moon arth ot drawn to scale aintaining athematical roficiency ind the value of. (ection.1) THTI OTIO The equations of the lines shown are y = + and y =. how that O O. y O eviewing what you learned in previous grades and lessons hapter imilarity