Similar Polygons Skill 36

Objective HSG-SRT.5: Students are responsible for identifying and applying similar polygons to solve problems.

Definitions Similar figures have the same shape but not necessarily the same size. When three or more ratios are equal, you can write an extended proportion. A scale factor is the ratio of corresponding linear measurements of two similar figures.

Similar Polygons B I C J A D H K Two polygons are similar polygons if corresponding angles are congruent and if the lengths of corresponding sides are proportional. B I C J A D H K

Example 1; Understanding Similarity Given: ∆𝑀𝑁𝑃~∆𝑆𝑅𝑇 a) What are the pairs of congruent angles? ∠𝑴≅∠𝑺 ∠𝑵≅∠𝑹 ∠𝑷≅∠𝑻 N M P R S T b) What is the extended proportion for the ratios of the lengths of corresponding sides? 𝑴𝑵 𝑺𝑹 = 𝑵𝑷 𝑹𝑻 = 𝑴𝑷 𝑺𝑻

Example 2; Determining Similarity Are the polygons similar? If they are, write a similarity statement and give the scale factor. If not explain why not.   a) 𝐽𝐾𝐿𝑀 and 𝑇𝑈𝑉𝑊 J K L M V U T W 16 6 12 14 24 Assume Corresponding angles are congruent. 𝑱𝑲 𝑻𝑼 = 𝟏𝟐 𝟔 𝑴𝑱 𝑾𝑻 = 𝟔 𝟔 𝑳𝑴 𝑽𝑾 = 𝟐𝟒 𝟏𝟒 𝑲𝑳 𝑼𝑽 = 𝟐𝟒 𝟏𝟔 JKLM is not similar to TUVW, because corresponding sides are not proportional.

Example 2; Determining Similarity Are the polygons similar? If they are, write a similarity statement and give the scale factor. If not explain why not. b) ∆𝐴𝐵𝐶 and ∆𝐷𝐸𝐹 B C A E F D Assume corresponding angles are congruent. 16 15 10 20 12 8 𝑨𝑩 𝑫𝑬 = 𝟏𝟓 𝟏𝟐 = 𝟓 𝟒 𝑩𝑪 𝑬𝑭 = 𝟐𝟎 𝟏𝟔 = 𝟓 𝟒 𝑨𝑪 𝑫𝑭 = 𝟏𝟎 𝟖 = 𝟓 𝟒 Yes, ∆ABC ~ ∆DEF with a scale factor of 5/4.

Example 3; Using Similar Polygons Consider 𝐴𝐵𝐶𝐷~𝐸𝐹𝐺𝐷 a) Find the value of x b) Find the value of y. 𝑭𝑮 𝑩𝑪 = 𝑬𝑫 𝑨𝑫 𝑬𝑭 𝑨𝑩 = 𝑬𝑫 𝑨𝑫 9 6 x 7.5 5 y B C D A F G E 𝒙 𝟕.𝟓 = 𝟔 𝟗 𝒚 𝟓 = 𝟔 𝟗 𝟗𝒙=𝟒𝟓 𝟗𝒚=𝟑𝟎 𝒙=𝟓 𝒚= 𝟏𝟎 𝟑

Example 4; Using Similar Polygons Consider 𝑊𝑋𝑌𝑍~𝑅𝑆𝑇𝑍 and 𝑊𝑋𝑌𝑍 is a parallelogram a) Find the value of x. b) Find the value of y. c) Find the value of z. 𝑹𝑺 𝑾𝑿 = 𝑺𝑻 𝑿𝒀 X Y Z W S T R zᵒ 60ᵒ x 20 24 yᵒ 16 𝟏𝟔 𝟐𝟒 = 𝒙 𝟐𝟎 𝟐𝟒𝒙=𝟑𝟐𝟎 Opposite Angles of Parallelogram are Congruent. 𝒚=𝟔𝟎 𝒙= 𝟒𝟎 𝟑 Consecutive Angles of Parallelogram are Supp. 𝒛=𝟏𝟐𝟎

Example 5; Using Similarity a) Your class is making a rectangular poster for a rally. The poster’s design is 6 in. high by 10 in. wide. The space allowed for the poster is 4ft. high by 8 ft. wide. What are the dimensions of the largest poster that will fit in the space? Height: 𝟒 𝒇𝒕.=𝟒𝟖 𝒊𝒏. Width: 𝟖 𝒇𝒕.=𝟗𝟔 𝒊𝒏. 𝟒𝟖𝒊𝒏÷𝟔𝒊𝒏=𝟖 𝟗𝟔𝒊𝒏÷𝟏𝟎𝒊𝒏=𝟗.𝟔 Biggest possible scale 𝟔 𝟒𝟖 = 𝟏𝟎 𝒙 𝟔𝒙=𝟒𝟖𝟎 𝒙=𝟖𝟎 The poster can be 48’’ by 80’’ or 4’ by 6’8’’

Example 5; Using Similarity b) Your class is making a rectangular poster for a rally. The poster’s design is 6 in. high by 10 in. wide. The space allowed for the poster is 3ft. high by 4 ft. wide. What are the dimensions of the largest poster that will fit in the space? Height: 𝟑 𝒇𝒕.=𝟑𝟔 𝒊𝒏. Width: 𝟒 𝒇𝒕.=𝟒𝟖 𝒊𝒏. 𝟑𝟔𝒊𝒏÷𝟔𝒊𝒏=𝟔 𝟒𝟖𝒊𝒏÷𝟏𝟎𝒊𝒏=𝟒.𝟖 𝟏𝟎 𝟒𝟖 = 𝟔 𝒙 Biggest possible scale 𝟒𝟎𝒙=𝟐𝟖𝟖 𝒙=𝟐𝟖.𝟖 The poster can be 28.8’’ by 48’’ or 2’4.8’’ by 4’

Example 6; Using a Scale Drawing A drawing of the Golden Gate Bridge in San Francisco is drawn to the scale of 1 cm to 200 m. the distance between the two towers is the main span, this is 6.4 cm in the drawing. What is the actual length of the main span of the bridge? 𝟏 𝒄𝒎 𝟐𝟎𝟎 𝒎 = 𝟔.𝟒 𝒄𝒎 𝒙 𝒎 𝒙=𝟐𝟎𝟎 𝟔.𝟒 𝒙=𝟏𝟐𝟖𝟎 𝒎

#36: Similar Polygons Questions? Summarize Notes Homework Video Quiz