Question

\(\triangle ABC\) and \(\triangle DEF\) are similar and their areas are 9 cm\(^2\) and 64 cm\(^2\) respectively. If DE = 5.1 cm then find AB.

Hint:

A common mistake is to forget to take the square root of the area ratio. Remember: Area ratio is the square of the side ratio, so the side ratio is the square root of the area ratio.

Answer 1

Step 1: Understanding the Concept:
This problem uses the theorem that states that 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:
If \(\triangle ABC \sim \triangle DEF\), then: \[ \frac{\text{Area}(\triangle ABC)}{\text{Area}(\triangle DEF)} = \left(\frac{AB}{DE}\right)^2 = \left(\frac{BC}{EF}\right)^2 = \left(\frac{AC}{DF}\right)^2 \]

Step 3: Detailed Explanation:
We are given:
Area(\(\triangle ABC\)) = 9 cm\(^2\)
Area(\(\triangle DEF\)) = 64 cm\(^2\)
DE = 5.1 cm
Since \(\triangle ABC \sim \triangle DEF\), AB and DE are corresponding sides.
Using the formula: \[ \frac{9}{64} = \left(\frac{AB}{5.1}\right)^2 \] To find the ratio of the sides, take the square root of both sides: \[ \sqrt{\frac{9}{64}} = \frac{AB}{5.1} \] \[ \frac{3}{8} = \frac{AB}{5.1} \] Now, solve for AB: \[ AB = \frac{3 \times 5.1}{8} \] \[ AB = \frac{15.3}{8} \] \[ AB = 1.9125 \text{ cm} \]

Step 4: Final Answer:
The length of side AB is 1.9125 cm.