Groundhog Day A 16 inch tall groundhog emerges on Groundhog Day near a tree and sees its shadow. The length of the groundhog’s shadow is 5 inches, and the length of the tree’s shadow is 35 inches. What is the height of the tree? In the Real World This problem can be solved using similar triangles, as you will see in Example 2. Using Proportions with Similar Polygons 10 6.

Groundhog Day A 16 inch tall groundhog emerges on Groundhog Day near a tree and sees its shadow. The length of the groundhog’s shadow is 5 inches, and the length of the tree’s shadow is 35 inches. What is the height of the tree? In the Real World This problem can be solved using similar triangles, as you will see in Example 2. Using Proportions with Similar Polygons Because the ratios of the lengths of corresponding sides are equal in similar polygons, you can write and solve proportions to find unknown lengths.

AD EH BC FG = Finding an Unknown Length EXAMPLE 1 Quadrilaterals ABCD and EFGH are similar. Find FG. Using Proportions with Similar Polygons cm 40 cm A B CD 20 cm x E F G H Use the ratios of the lengths of corresponding sides to write a proportion involving the unknown length. SOLUTION ANSWER The length of FG is 25 centimeters. ___ x = x = x 32 = AD EH BC FG = Write proportion involving FG. Substitute known values. Cross products property Divide each side by 32. Simplify x =

Using Proportions with Similar Polygons Indirect Measurement Because the sun’s rays hit objects that are perpendicular to the ground at the same angle, similar triangles are formed by objects and their shadows. You can use these similar triangles to find lengths that are difficult to measure directly.

16 16 Making an Indirect Measurement EXAMPLE 2 Using Proportions with Similar Polygons ANSWER The tree has a height of 112 inches, or 9 feet 4 inches. Groundhog Day You can use indirect measurement to find the height of the tree pictured on the right. Use the ratios of the lengths of corresponding sides to write a proportion involving the unknown height h. SOLUTION h = 112 h = 16 7 h = Substitute known values. Multiply each side by 16. Simplify fraction. Multiply. Length of tree’s shadow Length of groundhog’s shadow Height of tree Height of groundhog = h =