The areas of two similar triangles are 169 sq .cm and 121 sq. cm .If the longest side of larger triangle is 26 cm , then the length of the longest side of the other triangle is _____.

Options

A. 26 cm
B. 18 cm
C. 28 cm
D. 22 cm

Answer

To solve the problem, we need to find the length of the longest side of the smaller triangle given the areas of two similar triangles and the longest side of the larger triangle.

### Step-by-step Solution:

1. **Identify the Areas of the Triangles**:
- Area of the larger triangle (Triangle ABC) = 169 sq. cm
- Area of the smaller triangle (Triangle PQR) = 121 sq. cm

2. **Use the Property of Similar Triangles**:
- For similar triangles, the ratio of their areas is equal to the square of the ratio of their corresponding sides.
- Let the ratio of the sides of the larger triangle to the smaller triangle be \( k \).
- Thus, we can express the relationship as:
\[ \frac{\text{Area of Triangle ABC}}{\text{Area of Triangle PQR}} = k^2 \]
- Plugging in the areas:
\[ \frac{169}{121} = k^2 \]

3. **Calculate the Ratio of the Areas**:
- Simplifying the fraction:
\[ \frac{169}{121} = \left(\frac{13}{11}\right)^2 \]
- Therefore, we have:
\[ k = \frac{13}{11} \]

4. **Find the Longest Side of the Smaller Triangle**:
- The longest side of the larger triangle (AB) = 26 cm.
- Using the ratio of the sides:
\[ \frac{\text{Longest side of Triangle ABC}}{\text{Longest side of Triangle PQR}} = \frac{13}{11} \]
- Let the longest side of Triangle PQR be \( PQ \):
\[ \frac{26}{PQ} = \frac{13}{11} \]

5. **Cross-Multiply to Solve for PQ**:
- Cross-multiplying gives:
\[ 13 \cdot PQ = 26 \cdot 11 \]
- Simplifying:
\[ 13 \cdot PQ = 286 \]
- Dividing both sides by 13:
\[ PQ = \frac{286}{13} = 22 \]

6. **Conclusion**:
- The length of the longest side of the smaller triangle is **22 cm**.