Understanding Similar Triangles Area and Side Ratios

This question is about finding the ratio between corresponding sides of two similar triangles when the ratio of their areas is known. We need to determine the value of AB : PQ.

Key Concept: Area Ratio of Similar Triangles

A crucial theorem in geometry states that the ratio of the areas of two similar triangles is equal to the square of the ratio of their corresponding sides. If we have two similar triangles, $\triangle ABC$ and $\triangle PQR$, such that $\triangle ABC \sim \triangle PQR$, this relationship can be expressed mathematically as:

$$ \frac{\text{Area}(\triangle ABC)}{\text{Area}(\triangle PQR)} = \left( \frac{AB}{PQ} \right)^2 = \left( \frac{BC}{QR} \right)^2 = \left( \frac{AC}{PR} \right)^2 $$

This means the ratio of areas is the square of the ratio of any pair of corresponding sides (like AB and PQ).

Problem Breakdown

We are given the following details:

• The similarity statement: $\triangle ABC \sim \triangle PQR$.
• The ratio of the areas of these triangles: $$ \frac{\text{Area}(\triangle ABC)}{\text{Area}(\triangle PQR)} = \frac{25}{49} $$

Our goal is to find the ratio of the corresponding sides AB to PQ, denoted as AB : PQ.

Step-by-Step Calculation

Using the theorem about the areas of similar triangles, we can relate the given area ratio to the ratio of the sides AB and PQ:

$$ \left( \frac{AB}{PQ} \right)^2 = \frac{\text{Area}(\triangle ABC)}{\text{Area}(\triangle PQR)} $$

Now, we substitute the given value of the area ratio into the equation:

$$ \left( \frac{AB}{PQ} \right)^2 = \frac{25}{49} $$

To find the ratio $\frac{AB}{PQ}$, we need to take the square root of both sides of this equation. Remember that side lengths must be positive, so we consider the positive square root:

$$ \frac{AB}{PQ} = \sqrt{\frac{25}{49}} $$

Performing the square root calculation:

$$ \frac{AB}{PQ} = \frac{\sqrt{25}}{\sqrt{49}} $$

$$ \frac{AB}{PQ} = \frac{5}{7} $$

Thus, the ratio of the side AB to the side PQ is 5 to 7.

Conclusion

The value of the ratio AB : PQ is 5 : 7.