Question

The corresponding sides of two similar triangles are in the ratio 4 : 9. What will be the ratio of the areas of the triangles?

• A. 9 : 4
• B. 16 : 81
• C. 81 : 16
• D. 2 : 3

Hint:

To go from the ratio of sides to the ratio of areas, you square the numbers. To go from the ratio of areas to the ratio of sides, you take the square root. Don't mix them up.

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
This question uses the theorem that relates the ratio of the sides of similar triangles to the ratio of their areas.

Step 2: Key Formula or Approach:
If the ratio of the corresponding sides of two similar triangles is \(a : b\), then the ratio of their areas is \(a^2 : b^2\).
\[ \frac{\text{Area}_1}{\text{Area}_2} = \left(\frac{\text{Side}_1}{\text{Side}_2}\right)^2 \]

Step 3: Detailed Explanation:
We are given the ratio of the corresponding sides:
\[ \frac{\text{Side}_1}{\text{Side}_2} = \frac{4}{9} \] To find the ratio of the areas, we square this ratio:
\[ \frac{\text{Area}_1}{\text{Area}_2} = \left(\frac{4}{9}\right)^2 = \frac{4^2}{9^2} = \frac{16}{81} \] So, the ratio of the areas is 16 : 81.

Step 4: Final Answer:
The ratio of the areas of the triangles will be 16 : 81.