It is given that the triangles are similar, that is,
[tex]\triangle ABC\sim\triangle XYZ[/tex]
It is required to find the area of the larger triangle.
Recall that the scale factor, k of similar figures is the ratio of their corresponding sides:
[tex]k=\frac{AC}{XZ}[/tex]
Substitute AC=5 and XZ=15 into the equation:
[tex]k=\frac{5}{15}=\frac{1}{3}[/tex]
Hence, the scale factor is 1/3.
Recall that as per the Area of Similar Figures, the ratio of areas for two similar figures with a scale factor, k is:
[tex]k^2[/tex]
This implies that:
[tex]\frac{\text{area of }\triangle ABC}{\text{area of }\triangle XYZ}=k^2[/tex]
Substitute the values of the area of triangle ABC and the scale factor into the proportion:
[tex]\Rightarrow\frac{48}{\text{area of }\triangle XYZ}=(\frac{1}{3})^2[/tex]
Let the area of ΔXYZ be A, and solve for A in the equation:
[tex]\begin{gathered} \frac{48}{A}=\frac{1}{9} \\ \Rightarrow A=9\times48=432ft^2 \end{gathered}[/tex]
The required answer is 432 square ft.
The last choice is the answer.
