Understanding Similar Triangles and Area Ratio

This problem involves similar triangles. Similar triangles have the same shape but can be different sizes. A key property of similar triangles is that the ratio of their corresponding sides is constant. Another important property relates their areas to their corresponding sides.

Theorem: Area Ratio and Side Ratio of Similar Triangles

The theorem states that if two triangles are similar, then the ratio of the area of the first triangle to the area of the second triangle is equal to the square of the ratio of the length of any pair of corresponding sides.

Mathematically, if $\triangle ABC \sim \triangle PQR$, then:

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

Where \(AB\) and \(PQ\), \(BC\) and \(QR\), \(CA\) and \(RP\) are pairs of corresponding sides.

Applying the Theorem to the Problem

We are given that the areas of two similar triangles are in the ratio 196 ∶ 625. Let the two similar triangles be \(\triangle_1\) and \(\triangle_2\).

So, \(\frac{\text{Area}(\triangle_1)}{\text{Area}(\triangle_2)} = \frac{196}{625}\).

Let \(s_1\) and \(s_2\) be the lengths of corresponding sides of \(\triangle_1\) and \(\triangle_2\), respectively. According to the theorem:

\(\frac{\text{Area}(\triangle_1)}{\text{Area}(\triangle_2)} = \left(\frac{s_1}{s_2}\right)^2\)

Substituting the given area ratio:

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

Calculating the Ratio of Corresponding Sides

To find the ratio of the corresponding sides, \(\frac{s_1}{s_2}\), we need to take the square root of the ratio of the areas:

\(\frac{s_1}{s_2} = \sqrt{\frac{196}{625}}\)

We calculate the square root of the numerator and the denominator separately:

• \(\sqrt{196}\)
• \(\sqrt{625}\)

Let's find the square roots:

\(\sqrt{196} = 14\)

\(\sqrt{625} = 25\)

So, the ratio of the corresponding sides is:

\(\frac{s_1}{s_2} = \frac{14}{25}\)

This can also be written as a ratio 14 ∶ 25.

Comparing with the Options

The calculated ratio of corresponding sides is 14 ∶ 25. Let's compare this with the given options:

• Option 1: 14 ∶ 25
• Option 2: 13 ∶ 20
• Option 3: 14 ∶ 20
• Option 4: 13 ∶ 25

Our result matches Option 1.

Step-by-Step Solution Summary

Here is a summary of the steps taken to solve the problem:

1. Identify that the problem involves similar triangles and their area and side ratios.
2. Recall the theorem relating the ratio of areas to the square of the ratio of corresponding sides: \(\frac{\text{Area}_1}{\text{Area}_2} = \left(\frac{\text{side}_1}{\text{side}_2}\right)^2\).
3. Substitute the given area ratio: \(\frac{196}{625} = \left(\frac{\text{side}_1}{\text{side}_2}\right)^2\).
4. Take the square root of both sides to find the ratio of sides: \(\frac{\text{side}_1}{\text{side}_2} = \sqrt{\frac{196}{625}}\).
5. Calculate the square roots: \(\sqrt{196} = 14\) and \(\sqrt{625} = 25\).
6. Write the ratio of sides as \(\frac{14}{25}\) or 14 ∶ 25.

Revision Table: Similar Triangles Properties

Additional Information: Finding Square Roots

Finding the square root of a number is the inverse operation of squaring a number. For example, since \(14^2 = 14 \times 14 = 196\), the square root of 196 is 14. Similarly, since \(25^2 = 25 \times 25 = 625\), the square root of 625 is 25. When dealing with ratios of lengths (which are positive), we consider the positive square root.