Similar Triangles Area Ratio Concept

In similar triangles, the ratio of their areas is equivalent to the square of the ratio of their corresponding altitudes (or sides, medians, etc.).

Calculating Triangle Areas using Altitudes

This principle allows us to find the area of one similar triangle if we know the area of another and the ratio of their corresponding altitudes.

Problem Details for Similar Triangles

• Given altitudes: $h_1 = 4\text{ cm}$ and $h_2 = 6\text{ cm}$.
• Given area of the larger triangle: $A_2 = 36\text{ cm}^2$.
• We need to find the area of the other triangle, $A_1$.

Step-by-Step Area Calculation

1. Ratio of Altitudes: Determine the ratio between the altitudes. Since $4\text{ cm} < 6\text{ cm}$, $h_1$ corresponds to the smaller triangle and $h_2$ to the larger one. $ \frac{h_1}{h_2} = \frac{4\text{ cm}}{6\text{ cm}} = \frac{2}{3} $
2. Square of Altitude Ratio: Square this ratio to find the ratio of the areas. $ \left(\frac{h_1}{h_2}\right)^2 = \left(\frac{2}{3}\right)^2 = \frac{4}{9} $ This establishes the area relationship: $\frac{A_1}{A_2} = \frac{4}{9}$.
3. Calculate Unknown Area: Use the area ratio and the known area of the larger triangle ($A_2 = 36\text{ cm}^2$) to find the area of the other triangle ($A_1$). $ A_1 = \frac{4}{9} \times A_2 $ $ A_1 = \frac{4}{9} \times 36\text{ cm}^2 $ $ A_1 = 4 \times \frac{36}{9}\text{ cm}^2 $ $ A_1 = 4 \times 4\text{ cm}^2 $ $ A_1 = 16\text{ cm}^2 $

Final Area Calculation Result

The area of the other triangle is $16\text{ cm}^2$.