Calculating Area Ratio of Similar Triangles

Understanding the relationship between the sides and areas of similar triangles is a fundamental concept in geometry. When two triangles are similar, their corresponding angles are equal, and their corresponding sides are in proportion. There is a specific theorem that connects the ratio of their corresponding sides to the ratio of their areas.

Theorem on Areas of Similar Triangles

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

Let \(\triangle ABC\) and \(\triangle DEF\) be two similar triangles, denoted as \(\triangle ABC \sim \triangle DEF\). If \(s_1\) and \(s_2\) are the lengths of any pair of corresponding sides (for example, \(AB\) and \(DE\)), and \(A_1\) and \(A_2\) are their respective areas (Area(\(\triangle ABC\)) and Area(\(\triangle DEF\))), then the theorem can be expressed as:

\[ \frac{\text{Area}(\triangle ABC)}{\text{Area}(\triangle DEF)} = \left(\frac{AB}{DE}\right)^2 = \left(\frac{BC}{EF}\right)^2 = \left(\frac{AC}{DF}\right)^2 = \left(\frac{s_1}{s_2}\right)^2 \]

Applying the Theorem to the Given Problem

The question provides the ratio of the corresponding sides of two similar triangles. Let this ratio be \(s_1 : s_2\).

Given ratio of corresponding sides \( = \sqrt{5} : \sqrt{7} \)

This can be written as a fraction:

\[ \frac{s_1}{s_2} = \frac{\sqrt{5}}{\sqrt{7}} \]

We want to find the ratio of the areas of these two similar triangles. Let the areas be \(A_1\) and \(A_2\). According to the theorem on the areas of similar triangles, the ratio of their areas is the square of the ratio of their corresponding sides.

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

Substitute the given ratio of sides into this formula:

\[ \frac{A_1}{A_2} = \left(\frac{\sqrt{5}}{\sqrt{7}}\right)^2 \]

Now, calculate the square of the ratio:

\[ \left(\frac{\sqrt{5}}{\sqrt{7}}\right)^2 = \frac{(\sqrt{5})^2}{(\sqrt{7})^2} \]

The square of a square root is the number itself:

\[ (\sqrt{5})^2 = 5 \] \[ (\sqrt{7})^2 = 7 \]

So, the ratio of the areas is:

\[ \frac{A_1}{A_2} = \frac{5}{7} \]

The ratio of the area of the two triangles is \(5 : 7\).

Comparing with Options

Let's compare our calculated ratio with the given options:

• Option 1: \(\sqrt[3] {5} : \sqrt{7}\) - Incorrect.
• Option 2: 25 : 49 - Incorrect. This would be the ratio if the side ratio was 5:7.
• Option 3: \(\sqrt{5} : \sqrt{7}\) - Incorrect. This is the ratio of the sides, not the areas.
• Option 4: 5 : 7 - Correct. This matches our calculated ratio of the areas.

Conclusion

The ratio of the areas of the two similar triangles is \(5 : 7\).

Revision Table: Similar Triangles Ratios

Additional Information: Properties of Similar Triangles

Similar triangles have several important properties:

• Corresponding Angles: All pairs of corresponding angles are equal in measure.
• Corresponding Sides: The ratio of the lengths of corresponding sides is constant. This constant ratio is called the scale factor.
• Perimeters: The ratio of the perimeters of two similar triangles is equal to the ratio of their corresponding sides (the scale factor).
• Altitudes, Medians, Angle Bisectors: The ratio of corresponding altitudes, medians, or angle bisectors is also equal to the ratio of the corresponding sides (the scale factor).
• Areas: The ratio of the areas of two similar triangles is equal to the square of the ratio of their corresponding sides.

These properties are very useful for solving problems involving similar triangles, allowing us to find unknown lengths or areas if we know the scale factor or other ratios.