Question

amelia is 5 feet tall and casts a 4 - foot shadow. a tree next to her casts a 12 - foot shadow. the two triangles formed are similar because the angle to the sun is the same. choose two equations that can be used to find the height, h, of the tree.

Explanation:

Step1: Recall Similar Triangles Property

For similar triangles, the ratios of corresponding sides are equal. Amelia's height (5 ft) corresponds to the tree's height (\(h\) ft), and Amelia's shadow (4 ft) corresponds to the tree's shadow (12 ft). So one equation is \(\frac{5}{4}=\frac{h}{12}\).

Step2: Cross - Multiplication or Alternative Ratio

We can also set up the ratio by reversing the corresponding sides. The ratio of Amelia's shadow to her height is equal to the ratio of the tree's shadow to the tree's height. So another equation is \(\frac{4}{5}=\frac{12}{h}\).

Answer:

Two equations that can be used are \(\boldsymbol{\frac{5}{4}=\frac{h}{12}}\) and \(\boldsymbol{\frac{4}{5}=\frac{12}{h}}\) (or other equivalent forms like \(4h = 5\times12\) derived from cross - multiplying \(\frac{5}{4}=\frac{h}{12}\), and \(5\times12 = 4h\) is also equivalent to the second equation when cross - multiplied).