Problem 1 Finding Ratios in Similar Figures

The trapezoids below are similar. The ratio of the lengths of corresponding sides is 6 9 , 6 ninths , comma  or 2 3 . 2 thirds , .

1. What is the ratio (smaller to larger) of the perimeters?

   The ratio of the perimeters is the same as the ratio of corresponding sides, which is 2 3 . 2 thirds , .

2. What is the ratio (smaller to larger) of the areas?

   The ratio of the areas is the square of the ratio of corresponding sides, which is 2 2 3 2 , fraction 2 squared , over 3 squared end fraction . comma  or 4 9 . 4 ninths , .

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1. Two similar polygons have corresponding sides in the ratio 5 : 7.
   1. What is the ratio (larger to smaller) of their perimeters?
   2. What is the ratio (larger to smaller) of their areas?

Problem 2 Finding Areas Using Similar Figures

Multiple Choice The area of the smaller regular pentagon is about 27 . 5   cm 2 . 27 . 5 , cm squared , .  What is the best approximation for the area of the larger regular pentagon?

1. 11   cm 2 11 , cm squared
2. 69   cm 2 69 , cm squared
3. 172   cm 2 172 , cm squared
4. 275   cm 2 275 , cm squared

Regular pentagons are similar because all angles measure 108 and all sides in each pentagon are congruent. Here the ratio of corresponding side lengths is 4 10 , 4 tenths , comma  or 2 5 . 2 fifths , .  The ratio of the areas is 2 2 5 2 , or 4 25 . fraction 2 squared , over 5 squared end fraction . comma or , 4 twenty fifths , .

4 25 = 27.5 A Write a proportion using the ratio of the areas. 4 A = 687.5 Cross Products Property A = 687.5 4 Divide each side by  4   . A = 171.875 Simplify. table with 4 rows and 3 columns , row1 column 1 , 4 twenty fifths , column 2 equals , 27.5 over eh , column 3 cap write a proportion using the ratio of the areas. , row2 column 1 , 4 eh , column 2 equals , 687.5 , column 3 cap cross cap products cap property , row3 column 1 , eh , column 2 equals , 687.5 over 4 , column 3 cap divide each side by . 4 . , row4 column 1 , eh , column 2 equals , 171.875 , column 3 cap simplify. , end table

The area of the larger pentagon is about 172   cm 2 . 172 , cm squared , .  The correct answer is C.

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2. The scale factor of two similar parallelograms is 3 4 . 3 fourths , .  The area of the larger parallelogram is 96   in . 2 . 96 in , . squared , .  What is the area of the smaller parallelogram?