Given that two triangles are similar to each other and the ratio of their corresponding altitudes is 3:7, what is the ratio of their respective areas?

The correct option is 9:49.

Key Concept:
When two triangles are similar, the ratio of their areas is equal to the square of the ratio of their corresponding altitudes (heights), sides, medians, or perimeters.

Let the two similar triangles be triangle 1 and triangle 2.

The formula relating the ratio of their areas to the ratio of their corresponding altitudes is:

$\frac{\text{Area of Triangle 1}}{\text{Area of Triangle 2}}={\left(\frac{{\text{Altitude}}_1}{{\text{Altitude}}_2}\right)}^2$

Given in the problem, the ratio of their corresponding altitudes is:

$\frac{{\text{Altitude}}_1}{{\text{Altitude}}_2}=\frac{3}{7}$

Substituting this ratio into the area ratio formula gives:

$\frac{\text{Area of Triangle 1}}{\text{Area of Triangle 2}}={\left(\frac{3}{7}\right)}^2$

$\frac{\text{Area of Triangle 1}}{\text{Area of Triangle 2}}=\frac{3^2}{7^2}=\frac{9}{49}$

Therefore, the ratio of their respective areas is 9:49.