360 sq. cm and 250 sq. cm are the area of two similar triangles. If the length of one of the sides of the first triangle be 8 cm, then the length of the corresponding side of the second triangle is

Options

A. `6(1)/(5)cm`
B. `6(1)/(3)cm`
C. `6(2)/(3)cm`
D. 6 cm

Answer

To find the length of the corresponding side of the second triangle, we can use the property of similar triangles, which states that the ratio of the areas of two similar triangles is equal to the square of the ratio of their corresponding sides.

### Step-by-Step Solution:

1. **Identify the areas of the triangles**:
- Area of Triangle 1 (A1) = 360 sq. cm
- Area of Triangle 2 (A2) = 250 sq. cm

2. **Write the ratio of the areas**:
\[ \frac{A1}{A2} = \frac{360}{250} \]

3. **Simplify the ratio**:
\[ \frac{360}{250} = \frac{36}{25} \quad \text{(dividing both by 10)} \]

4. **Set up the ratio of the sides**:
Let the length of the corresponding side of Triangle 2 be \( x \). The length of the side of Triangle 1 is given as 8 cm.
\[ \frac{A1}{A2} = \frac{(side1)^2}{(side2)^2} \implies \frac{36}{25} = \frac{8^2}{x^2} \]

5. **Substitute the values**:
\[ \frac{36}{25} = \frac{64}{x^2} \]

6. **Cross-multiply to solve for \( x^2 \)**:
\[ 36x^2 = 25 \times 64 \]
\[ 36x^2 = 1600 \]

7. **Divide both sides by 36**:
\[ x^2 = \frac{1600}{36} \]

8. **Simplify \( x^2 \)**:
\[ x^2 = \frac{400}{9} \]

9. **Take the square root of both sides**:
\[ x = \sqrt{\frac{400}{9}} = \frac{\sqrt{400}}{\sqrt{9}} = \frac{20}{3} \]

10. **Convert to decimal or mixed number**:
\[ x = 6 \frac{2}{3} \text{ cm} \]

### Final Answer:
The length of the corresponding side of the second triangle is \( 6 \frac{2}{3} \) cm.

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