Question

complete parts a and b.
a. if triangle abc is similar to triangle def and \\( \frac { a b } { d e } = \frac { 1 } { 7 } \\), what is the ratio of the heights of the triangles?
b. what is the ratio of the areas of the two triangles?
a. the ratio of the height of triangle abc to the height of triangle def is (type an integer or a simplified fraction.)

Explanation:

Step1: Recall the property of similar triangles for heights

For similar triangles, the ratio of corresponding heights is equal to the ratio of corresponding sides.
Given \(\frac{AB}{DE}=\frac{1}{7}\), so the ratio of the height of triangle \(ABC\) to the height of triangle \(DEF\) is \(\frac{1}{7}\).

Step2: Recall the property of similar triangles for areas

For similar triangles, the ratio of areas is equal to the square of the ratio of corresponding sides.
Since \(\frac{AB}{DE}=\frac{1}{7}\), the ratio of the area of triangle \(ABC\) to the area of triangle \(DEF\) is \((\frac{1}{7})^2=\frac{1}{49}\).

Answer:

a. \(\frac{1}{7}\)
b. \(\frac{1}{49}\)