Understanding the Area Ratio of Similar Triangles

When two triangles are similar, their corresponding sides are in proportion, and their corresponding angles are equal. There is a specific relationship between the ratio of their sides and the ratio of their areas.

Relationship Between Side Ratio and Area Ratio

The theorem relating the sides and areas of similar triangles states that the ratio of the areas of two similar triangles is equal to the square of the ratio of their corresponding sides.

Let $\triangle ABC$ be similar to $\triangle PQR$. If the ratio of their corresponding sides is $\frac{AB}{PQ} = \frac{BC}{QR} = \frac{CA}{RP} = k$, then the ratio of their areas is:

$\frac{\text{Area}(\triangle ABC)}{\text{Area}(\triangle PQR)} = k^2 = \left(\frac{AB}{PQ}\right)^2 = \left(\frac{BC}{QR}\right)^2 = \left(\frac{CA}{RP}\right)^2$.

Calculating the Area Ratio

In this problem, the sides of two similar triangles are in the ratio 4 ∶ 9.

Let the ratio of the sides be $s_1 : s_2 = 4 : 9$. This can be written as $\frac{s_1}{s_2} = \frac{4}{9}$.

According to the theorem, the ratio of their areas ($A_1 : A_2$) is the square of the ratio of their sides:

$\frac{A_1}{A_2} = \left(\frac{s_1}{s_2}\right)^2$

Substitute the given side ratio:

$\frac{A_1}{A_2} = \left(\frac{4}{9}\right)^2$

Calculate the square:

$\frac{A_1}{A_2} = \frac{4^2}{9^2} = \frac{16}{81}$

Thus, the ratio of the areas of the two similar triangles is 16 ∶ 81.

Analyzing the Options

• 2 ∶ 3: This is the square root of a possible area ratio, or unrelated.
• 4 ∶ 9: This is the given ratio of the sides, not the ratio of the areas.
• 81 ∶ 16: This is the reciprocal of the correct area ratio. The order is important. Since the sides are in the ratio 4:9 (smaller to larger), the areas will be in the ratio $4^2 : 9^2$ (smaller area to larger area), which is 16:81.
• 16 ∶ 81: This is the square of the side ratio ($4^2 : 9^2$), which is correct for the ratio of areas of similar triangles.

Therefore, the area of these triangles are in the ratio 16 ∶ 81.