Question

\(\triangle ABC \sim \triangle DEF\) and BC = 3 cm, EF = 4 cm. If the area of \(\triangle ABC\) is 54 cm\(^2\), then the area of \(\triangle DEF\) is

• A. 56 cm\(^2\)
• B. 96 cm\(^2\)
• C. 196 cm\(^2\)
• D. 49 cm\(^2\)

Hint:

When setting up the ratio, make sure to be consistent. If you put the area of triangle ABC in the numerator, you must also put its corresponding side (BC) in the numerator of the side ratio.

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
The ratio of the areas of two similar triangles is equal to the square of the ratio of their corresponding sides.

Step 2: Key Formula or Approach:
\[ \frac{\text{Area}(\triangle ABC)}{\text{Area}(\triangle DEF)} = \left(\frac{BC}{EF}\right)^2 \]

Step 3: Detailed Explanation:
We are given:
Area(\(\triangle ABC\)) = 54 cm\(^2\)
BC = 3 cm
EF = 4 cm
Let the Area(\(\triangle DEF\)) be \(x\).
Substitute the values into the formula:
\[ \frac{54}{x} = \left(\frac{3}{4}\right)^2 \] \[ \frac{54}{x} = \frac{9}{16} \] Now, solve for \(x\):
\[ 9x = 54 \times 16 \] \[ x = \frac{54 \times 16}{9} \] Since \(54/9 = 6\):
\[ x = 6 \times 16 = 96 \] The area of \(\triangle DEF\) is 96 cm\(^2\).

Step 4: Final Answer:
The area of \(\triangle DEF\) is 96 cm\(^2\).