Corresponding sides of two similar triangles are in the ratio \(2:3\). If the area of the smaller is \(48\,\text{cm}^2\), find the area of the larger triangle.
108 cm²
Step 1: In similar triangles, the ratio of their areas is equal to the square of the ratio of their corresponding sides.
Here, side ratio = \(2:3\).
So, area ratio = \( (2:3)^2 = (2^2 : 3^2) = 4:9 \).
Step 2: Let the area of the larger triangle be \(A_{\text{large}}\).
Then, according to the ratio:
\( \dfrac{\text{Area of smaller}}{\text{Area of larger}} = \dfrac{4}{9} \).
That is, \( \dfrac{48}{A_{\text{large}}} = \dfrac{4}{9} \).
Step 3: Cross multiply:
\( 48 \times 9 = 4 \times A_{\text{large}} \).
\( 432 = 4A_{\text{large}} \).
Step 4: Divide both sides by 4:
\( A_{\text{large}} = \dfrac{432}{4} = 108 \).
Final Answer: The area of the larger triangle is 108 cm².