Question

9. Given similar triangles ABC and PQR with AB = 4 and PQ = 7.

Answer 1

Given:

Similar triangles ABC and PQR with AB = 4 and PQ = 7.

To find:

The scale factor, ratio of perimeters, and ratio of areas.

Solution:

In similar figures the scale factor is the ratio of side or image and corresponding side of original figure.

In similar triangles ABC and PQR, the scale factor is

$k=\dfrac{PQ}{AB}$

$k=\dfrac{7}{4}$

The scale factor of triangle ABC to triangle PQR is $\dfrac{7}{4}$.

The ratio of the perimeters of similar triangles is equal to the ratio of their corresponding sides.

$\dfrac{\text{Perimeter of }\Delta ABC}{\text{Perimeter of }\Delta PQR}=\dfrac{AB}{PQ}$

$\dfrac{\text{Perimeter of }\Delta ABC}{\text{Perimeter of }\Delta PQR}=\dfrac{4}{7}$

$\dfrac{\text{Perimeter of }\Delta ABC}{\text{Perimeter of }\Delta PQR}=4:7$

The ratio of the perimeters is 4:7.

The ratio of the areas of similar triangles is equal to the ratio of squares of their corresponding sides.

$\dfrac{\text{Area of }\Delta ABC}{\text{Area of }\Delta PQR}=\dfrac{AB^2}{PQ^2}$

$\dfrac{\text{Area of }\Delta ABC}{\text{Area of }\Delta PQR}=\dfrac{4^2}{7^2}$

$\dfrac{\text{Area of }\Delta ABC}{\text{Area of }\Delta PQR}=\dfrac{16}{49}$

$\dfrac{\text{Area of }\Delta ABC}{\text{Area of }\Delta PQR}=16:49$

Therefore, the ratio of the areas is 16:49.