Question : $\triangle$ABC is similar to $\triangle$PQR. The length of AB is 16 cm and the length of the corresponding side PQ is 9 cm. If the area of $\triangle$ABC is 1024 sq. cm, what is the area of $\triangle$PQR?
Option 1: 768 sq. cm
Option 2: 32 sq. cm
Option 3: 324 sq. cm
Option 4: 128 sq. cm
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Correct Answer: 324 sq. cm
Solution :
In similar triangles, the ratio of their areas is the square of the ratio of their corresponding sides.
Given that $\triangle$ABC is similar to $\triangle$PQR, the length of AB is 16 cm, the length of the corresponding side PQ is 9 cm, and the area of $\triangle$ABC is 1024 sq. cm,
Area of $\triangle {PQR}$ = Area of $\triangle ABC\times \left(\frac{PQ}{AB}\right)^2$
Area of $\triangle {PQR}$ = $1024 \times \left(\frac{9}{16}\right)^2 = 1024 \times \left(\frac{81}{256}\right) = 324 \text{ sq. cm}$
Hence, the correct answer is 324 sq. cm.
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