Similar Polygons

Transcription

1 imilar olgons oal Identif similar polgons. e Words similar polgons scale factor In geometr, two figures that have the same shape are called similar. wo polgons are similar polgons if corresponding angles are congruent and corresponding side lengths are proportional. In the diagram, is similar to. he smbol indicates similarit. o, ou can write. When ou refer to similar polgons, list their corresponding vertices in the same order. orresponding ngles atios of orresponding ides tud t lp a c a a c a a c a a c a VOUY I Q U is called a similarit statement. X 1 Use imilarit tatements Q U. a. ist all pairs of congruent angles. b. Write the ratios of the corresponding sides in a statement of proportionalit. c. heck that the ratios of corresponding sides are equal. U olution a. a ca, a ca, and aq cau. b. U Q U Q c. 5, U Q 1 5, and U Q he ratios of corresponding sides are all equal to imilar olgons 35

2 cale actor If two polgons are similar, then the ratio of the lengths of two corresponding sides is called the. scale factor scale factor of to Itudent elp I ZO.O O X ore eamples at classzone.com X 2 etermine Whether olgons are imilar etermine whether the triangles are similar. If the are similar, write a similarit statement and find the scale factor of igure to igure. olution 1 heck whether the corresponding angles are congruent. rom the diagram, ou can see that a ca, a ca, and a ca. herefore, the corresponding angles are congruent. 2 heck whether the corresponding side lengths are proportional ll three ratios are equal, so the corresponding side lengths are proportional. W definition, the triangles are similar.. he scale factor of igure to igure is 3. etermine Whether olgons are imilar etermine whether the polgons are similar. If the are similar, write a similarit statement and find the scale factor of igure to igure. 1. Y 2. X Z 1 V U 3 hapter 7 imilarit

3 X 3 Use imilar olgons. ind the value of. tud t lp UY I You can find using other proportions that include. nother proportion ou can use is. olution ecause the triangles are similar, the corresponding side lengths are proportional. o find the value of, ou can use the following proportion. 1 5 Write proportion. p p ross product propert 0 ultipl ivide each side b. 5 ubstitute for, for, for, and for. implif. X erimeters of imilar olgons he outlines of a pool and the patio around the pool are similar rectangles. a. ind the ratio of the length of the patio to the length of the pool. b. ind the ratio of the perimeter of the patio to the perimeter of the pool. olution 32 ft ft ft 2 ft a. he ratio of the length of the patio to the length of the pool is l ength of patio feet 3 length of pool 32 feet b. he perimeter of the patio is 2(2) 2() 1 feet. he perimeter of the pool is 2() 2(32) feet. he ratio of the perimeter of the patio to the perimeter of the pool is perimeter of patio 1 feet f eet 1 3 perimeter of pool imilar olgons 37

4 In ample on the previous page, notice that the ratio of the perimeters of the similar figures is equal to the ratio of the side lengths. his observation is generalized in the following theorem. O 7.1 erimeters of imilar olgons Words If two polgons are similar, then the ratio of their perimeters is equal to the ratio of their corresponding side lengths. mbols If, then. Use imilar olgons In the diagram, Q U. 3. ind the value of.. ind the ratio of the perimeter of U to the perimeter of Q. 1 7 U 7.2 ercises uided ractice Vocabular heck 1. If two triangles are congruent, must the be similar? plain. 2. If two triangles are similar, must the be congruent? plain. kill heck In ercises 3,. 3. ist all pairs of congruent angles.. Write the ratios of the corresponding sides in a statement of proportionalit ind the scale factor of to.. ind the value of. 7. re the two rectangles shown at the right similar? plain our reasoning. 7 3 hapter 7 imilarit

5 ractice and pplications tra ractice ee p. 7. Using imilarit tatements ist all pairs of congruent angles. hen write the ratios of the corresponding sides in a statement of proportionalit.. Q. QU U. rror nalsis. student was asked to list all pairs of congruent angles and write the ratios of the corresponding sides in a statement of proportionalit. op the diagram and correct the student s errors. a ca, a ca, a ca etermining imilarit etermine whether the polgons are similar. If the are similar, write a similarit statement and find the scale factor of igure to igure. omework elp ample 1: s., 22, 23 ample 2: s. 11, 2 ample 3: s , 2, 25 ample : s. 2, V 5 U 5. U. X.75 W V 3 Z Z W Z 7 Y X Y.5 Y 7.2 imilar olgons 3

6 Using imilar olgons he two polgons are similar. ind the values of and areers ural askia iebrand designed a mural for the lace des rts subwa station in ontreal, anada. 5 in. b 21 in. sketch of the center section of the mural is shown below. he scale factor is about 3.5. What are the dimensions of this section of the mural in inches? In feet? I askia iebrand, above, designs and creates mosaic murals using ceramic tiles. er work is displaed in private homes and offices, as well as public buildings. areer inks ZO.O 370 hapter 7 imilarit Using imilar olgons In ercises 22 2, use the diagram below, where. 22. ind ma. 23. ind ma. 2. ind the value of. 25. ind the value of. 2. ind the ratio of the perimeter of to the perimeter of. 27. erimeters of imilar olgons Q is similar to XYZ. he ratio of one side of XYZ to the corresponding side of Q is : 7. What is the ratio of the perimeter of XYZ to the perimeter of Q? 2. hallenge is similar to U. he ratio of to is 5 to 2. he perimeter of U is 35 feet. ind the perimeter of. 0 1

7 in. echnolog 25 in. in. 2. elevision creens he aspect ratio of a television screen is the length-to-width ratio of the screen. standard definition television has an aspect ratio of : 3. high definition projection television has an aspect ratio of :. re the television screens similar rectangles? ogical easoning X re two isosceles trapezoids alwas, sometimes, or never similar? VIIO have sizes that are based on the length of their diagonal. he aspect ratio of a 25-inch standard definition television screen is in. : in., or : 3. olution It is possible to sketch two similar isosceles trapezoids. It is also possible to sketch two isosceles trapezoids that are not similar W herefore, two isosceles trapezoids are sometimes similar. ogical easoning re the polgons alwas, sometimes, or never similar? 30. wo isosceles triangles 31. wo rhombuses 32. wo equilateral triangles 33. right and an isosceles triangle tandardized est ractice 3. ultiple hoice. Which statement ma be false? a ca a ca ied eview howing riangles are ongruent oes the diagram give enough information to show that the triangles are congruent? If so, state the postulate or theorem ou would use. (essons 5.3, 5.) U lgebra kills Writing quivalent ractions Write two equivalent fractions. (kills eview, p. 5) imilar olgons 371