Finding the Ratio of Corresponding Sides of Similar Triangles

The question asks us to find the ratio of the corresponding sides of two similar triangles, given that the ratio of their areas is 196 : 625.

When two triangles are similar, there is a special relationship between the ratio of their areas and the ratio of their corresponding sides. This relationship is a fundamental theorem in geometry concerning similar figures.

Understanding the Similar Triangles Area Theorem

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 ($\triangle ABC \sim \triangle DEF$). Let their areas be Area($\triangle ABC$) and Area($\triangle DEF$). Let $AB$ and $DE$ be a pair of corresponding sides, $BC$ and $EF$ another pair, and $AC$ and $DF$ the third pair.

According to the theorem:

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

This means if the ratio of corresponding sides is $s_1 : s_2$, then the ratio of their areas is $s_1^2 : s_2^2$. Conversely, if the ratio of areas is $A_1 : A_2$, then the ratio of corresponding sides is $\sqrt{A_1} : \sqrt{A_2}$.

Applying the Theorem to Find Side Ratio

Given the ratio of the areas of the two similar triangles is 196 : 625. Let this ratio be $\frac{\text{Area}_1}{\text{Area}_2} = \frac{196}{625}$.

Let the ratio of their corresponding sides be $\frac{s_1}{s_2}$.

Using the theorem, we have:

$$ \frac{\text{Area}_1}{\text{Area}_2} = \left(\frac{s_1}{s_2}\right)^2 $$

Substitute the given area ratio:

$$ \frac{196}{625} = \left(\frac{s_1}{s_2}\right)^2 $$

To find the ratio of the sides $\frac{s_1}{s_2}$, we need to take the square root of both sides of the equation:

$$ \frac{s_1}{s_2} = \sqrt{\frac{196}{625}} $$

Now, we calculate the square roots of the numerator and the denominator:

• $\sqrt{196}$ is the number that, when multiplied by itself, gives 196. This number is 14 ($14 \times 14 = 196$).
• $\sqrt{625}$ is the number that, when multiplied by itself, gives 625. This number is 25 ($25 \times 25 = 625$).

So, the ratio of the corresponding sides is:

$$ \frac{s_1}{s_2} = \frac{14}{25} $$

The ratio of the corresponding sides is 14 : 25.

Comparing with Given Options

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

Our calculated ratio, 14 : 25, matches Option 1.

Conclusion

Based on the theorem relating the areas and corresponding sides of similar triangles, the ratio of the corresponding sides is the square root of the ratio of their areas. Given the area ratio 196 : 625, the side ratio is $\sqrt{196} : \sqrt{625}$, which simplifies to 14 : 25.

Revision Table: Similar Triangle Ratios

Additional Information on Similar Triangles

Similar triangles are triangles that have the same shape but different sizes. Their corresponding angles are equal, and their corresponding sides are in proportion.

• Corresponding Angles: If $\triangle ABC \sim \triangle DEF$, then $\angle A = \angle D$, $\angle B = \angle E$, and $\angle C = \angle F$.
• Corresponding Sides: The ratio of the lengths of corresponding sides is constant. This constant ratio is sometimes called the scale factor. If $\triangle ABC \sim \triangle DEF$, then $\frac{AB}{DE} = \frac{BC}{EF} = \frac{AC}{DF} = k$, where $k$ is the scale factor.
• Properties derived from similarity: As seen in the revision table, the ratio of corresponding altitudes, medians, angle bisectors, and perimeters are all equal to the ratio of corresponding sides (the scale factor). The ratio of areas is the square of this scale factor.

Understanding these properties is crucial for solving problems involving similar triangles in geometry.