Question

If the ratio of corresponding sides of two similar triangles is \(2:7\), then the ratio of the areas of the triangles is _____.

• A. \(\sqrt{2} : \sqrt{7}\)
• B. \(2 : 7\)
• C. \(7 : 2\)
• D. \(4 : 49\)

Hint:

For similar figures:
• Side ratio \(= k\)
• Area ratio \(= k^2\)
• Volume ratio \(= k^3\) Always remember the powers according to dimensions.

Answer 1

The Correct Option is D

Concept: For similar triangles:
• ratio of corresponding sides gives the scale factor,
• ratio of areas equals the square of the ratio of corresponding sides. Mathematically: \[ \frac{\text{Area}_1}{\text{Area}_2} = \left( \frac{\text{Corresponding Side}_1}{\text{Corresponding Side}_2} \right)^2 \]

Step 1: Write the given ratio of corresponding sides. \[ \frac{s_1}{s_2} = \frac{2}{7} \] where:
• \(s_1\) = side of first triangle
• \(s_2\) = side of second triangle

Step 2: Use area ratio property of similar triangles. \[ \frac{\text{Area}_1}{\text{Area}_2} = \left( \frac{2}{7} \right)^2 \]

Step 3: Square numerator and denominator separately. \[ \left( \frac{2}{7} \right)^2 = \frac{2^2}{7^2} = \frac{4}{49} \]

Step 4: Write the ratio form. \[ \frac{4}{49} = 4:49 \]

Step 5: Understand why squaring is needed. Lengths are one-dimensional quantities. Areas are two-dimensional quantities. So whenever lengths scale by a factor \(k\), areas scale by: \[ k^2 \] That is why the side ratio must always be squared. Final Answer: \[ \boxed{4:49} \]