Complete parts a and b. a. If triangle \( A B C \) is similar to triangle \( D E F \) and \( \frac{A B}{D E}=\frac{1}{3} \), 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 \( A B C \) to the height of triangle \( D E F \) is \( \frac{1}{8} \). (Type an integer or a simplified fraction.) b. The ratio of the area of traingle \( A B C \) to the area of triangle \( D E F \) is \( \frac{1}{64} \). (Type an integer or a simplified fraction.)

Given that triangle \( A B C \) is similar to triangle \( D E F \) and that the ratio of their corresponding sides is \( \frac{A B}{D E}=\frac{1}{3} \): a. Since the height of similar triangles is in the same ratio as their corresponding sides, the height ratio will also be \( \frac{1}{3} \). Therefore, the ratio of the height of triangle \( A B C \) to the height of triangle \( D E F \) is \( \frac{1}{3} \). b. The area of similar triangles is proportional to the square of the ratio of their corresponding sides. Thus, the ratio of the areas will be \( \left(\frac{1}{3}\right)^2 = \frac{1}{9} \). Therefore, the ratio of the area of triangle \( A B C \) to the area of triangle \( D E F \) is \( \frac{1}{9} \).