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
[tex]k=\dfrac{PQ}{AB}[/tex]
[tex]k=\dfrac{7}{4}[/tex]
The scale factor of triangle ABC to triangle PQR is [tex]\dfrac{7}{4}[/tex].
The ratio of the perimeters of similar triangles is equal to the ratio of their corresponding sides.
[tex]\dfrac{\text{Perimeter of }\Delta ABC}{\text{Perimeter of }\Delta PQR}=\dfrac{AB}{PQ}[/tex]
[tex]\dfrac{\text{Perimeter of }\Delta ABC}{\text{Perimeter of }\Delta PQR}=\dfrac{4}{7}[/tex]
[tex]\dfrac{\text{Perimeter of }\Delta ABC}{\text{Perimeter of }\Delta PQR}=4:7[/tex]
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.
[tex]\dfrac{\text{Area of }\Delta ABC}{\text{Area of }\Delta PQR}=\dfrac{AB^2}{PQ^2}[/tex]
[tex]\dfrac{\text{Area of }\Delta ABC}{\text{Area of }\Delta PQR}=\dfrac{4^2}{7^2}[/tex]
[tex]\dfrac{\text{Area of }\Delta ABC}{\text{Area of }\Delta PQR}=\dfrac{16}{49}[/tex]
[tex]\dfrac{\text{Area of }\Delta ABC}{\text{Area of }\Delta PQR}=16:49[/tex]
Therefore, the ratio of the areas is 16:49.
