The legs of a right- angled triangle area in the ratio `5:12.` If its ara is 120 sq. m, find the legs.

Answer

To solve the problem step by step, we will follow the logic used in the video transcript.

### Step-by-Step Solution:

1. **Understand the ratio of the legs**:
The legs of the right-angled triangle are in the ratio of 5:12. Let's denote the legs as:
- Base (leg1) = 5x
- Height (leg2) = 12x

2. **Use the formula for the area of a triangle**:
The area \( A \) of a triangle is given by the formula:
\[ A = \frac{1}{2} \times \text{Base} \times \text{Height} \]
Here, the area is given as 120 sq. m. Therefore, we can write:
\[ 120 = \frac{1}{2} \times (5x) \times (12x) \]

3. **Simplify the equation**:
Multiply both sides by 2 to eliminate the fraction:
\[ 240 = 5x \times 12x \]
This simplifies to:
\[ 240 = 60x^2 \]

4. **Solve for \( x^2 \)**:
Divide both sides by 60:
\[ x^2 = \frac{240}{60} = 4 \]

5. **Find the value of \( x \)**:
Taking the square root of both sides gives:
\[ x = \sqrt{4} = 2 \]

6. **Calculate the lengths of the legs**:
Now, substitute \( x \) back into the expressions for the legs:
- Base (leg1) = \( 5x = 5 \times 2 = 10 \) meters
- Height (leg2) = \( 12x = 12 \times 2 = 24 \) meters

### Final Answer:
The lengths of the legs of the triangle are:
- Base = 10 meters
- Height = 24 meters

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