Question

In \(\triangle \text{ABC}\) and \(\triangle \text{DEF}\), \(\frac{\text{AB}}{\text{DE}} = \frac{\text{AC}}{\text{DF}} = \frac{\text{BC}}{\text{EF}} = \frac{3}{4}\). Then Area (\(\triangle \text{ABC}\)) : Area (\(\triangle \text{DEF}\)) is equal to :

• A. \(3:4\)
• B. \(16:9\)
• C. \(9:16\)
• D. \(27:64\)

Hint:

Key theorem: If two triangles are similar, the ratio of their areas is the square of the ratio of their corresponding sides. Given: Ratio of sides = \(\frac{3}{4}\). Ratio of areas = \((\text{Ratio of sides})^2 = \left(\frac{3}{4}\right)^2 = \frac{9}{16}\). So the ratio of areas is \(9:16\).

Answer 1

The Correct Option is C

Concept: If two triangles are similar, then the ratio of their areas is equal to the square of the ratio of their corresponding sides. The given condition \(\frac{\text{AB}}{\text{DE}} = \frac{\text{AC}}{\text{DF}} = \frac{\text{BC}}{\text{EF}} = \frac{3}{4}\) means that \(\triangle \text{ABC}\) is similar to \(\triangle \text{DEF}\) (by SSS similarity criterion, as the ratios of all corresponding sides are equal). Step 1: Identify the ratio of corresponding sides The ratio of corresponding sides is given as \(\frac{3}{4}\). Let this ratio be \(k\). So, \(k = \frac{3}{4}\). Step 2: Apply the theorem for the ratio of areas of similar triangles The theorem states: \[ \frac{\text{Area}(\triangle \text{ABC})}{\text{Area}(\triangle \text{DEF})} = \left(\frac{\text{AB}}{\text{DE}}\right)^2 = \left(\frac{\text{AC}}{\text{DF}}\right)^2 = \left(\frac{\text{BC}}{\text{EF}}\right)^2 = k^2 \] Step 3: Calculate the ratio of the areas Substitute the value of the ratio of sides \(k = \frac{3}{4}\) into the formula: \[ \frac{\text{Area}(\triangle \text{ABC})}{\text{Area}(\triangle \text{DEF})} = \left(\frac{3}{4}\right)^2 \] \[ \frac{\text{Area}(\triangle \text{ABC})}{\text{Area}(\triangle \text{DEF})} = \frac{3^2}{4^2} = \frac{9}{16} \] So, Area (\(\triangle \text{ABC}\)) : Area (\(\triangle \text{DEF}\)) = \(9:16\). This matches option (3).