Understanding Ratios in Similar Triangles

When we talk about similar triangles, we are referring to two triangles that have the same shape but can be different in size. This means their corresponding angles are equal, and their corresponding sides are proportional.

A key property of similar triangles relates the ratio of their corresponding sides to the ratio of other corresponding linear measurements, such as altitudes, medians, and angle bisectors.

Relationship between Sides and Altitudes in Similar Triangles

For any two similar triangles, the ratio of their corresponding altitudes is always equal to the ratio of their corresponding sides.

Let's consider two similar triangles, $\triangle ABC$ and $\triangle PQR$. If $\triangle ABC \sim \triangle PQR$, then the ratio of their corresponding sides is constant:

$$ \frac{AB}{PQ} = \frac{BC}{QR} = \frac{CA}{RP} $$

Now, if we draw corresponding altitudes, say $AD$ in $\triangle ABC$ (from $A$ to $BC$) and $PS$ in $\triangle PQR$ (from $P$ to $QR$), the property states that:

$$ \frac{AD}{PS} = \frac{AB}{PQ} = \frac{BC}{QR} = \frac{CA}{RP} $$

This is a fundamental property derived from the definition of similar triangles and trigonometric ratios or by considering smaller similar triangles formed by the altitudes.

Applying the Given Ratio

The question provides that the ratio of the corresponding sides of two similar triangles is 2 ∶ 3. We can write this ratio as a fraction:

$$ \text{Ratio of sides} = \frac{\text{Side 1}}{\text{Side 2}} = \frac{2}{3} $$

Based on the property discussed above, the ratio of their corresponding altitudes is equal to the ratio of their corresponding sides.

$$ \text{Ratio of altitudes} = \text{Ratio of sides} $$

Therefore, the ratio of their corresponding altitude is also 2 ∶ 3.

Conclusion

If the ratio of corresponding sides of 2 similar triangles is 2 ∶ 3, then the ratio of their corresponding altitude is also 2 ∶ 3.

Let's look at the options:

• 1 ∶ 3
• 4 ∶ 9
• 3 ∶ 2
• 2 ∶ 3

The ratio 2 ∶ 3 matches the calculated ratio of corresponding altitudes.