The area of a right angled triangle is 24 `cm^(2)` and the length of its hypotenuse is 10 cm. The length of the shorter leg is :

Options

A. 5 cm
B. 4 cm
C. 6 cm
D. 3 cm

Answer

To find the length of the shorter leg of the right-angled triangle, we can use the area and the hypotenuse given in the problem. Let's denote the shorter leg as \( x \) and the longer leg as \( y \).

### Step-by-Step Solution:

1. **Use the Area Formula**:
The area \( A \) of a right-angled triangle can be expressed as:
\[ A = \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times x \times y \]
Given that the area is \( 24 \, \text{cm}^2 \), we can set up the equation:
\[ \frac{1}{2} \times x \times y = 24 \]
Multiplying both sides by 2 gives:
\[ x \times y = 48 \quad \text{(Equation 1)} \]

2. **Use the Pythagorean Theorem**:
According to the Pythagorean theorem, for a right-angled triangle:
\[ x^2 + y^2 = \text{hypotenuse}^2 \]
Given that the hypotenuse is \( 10 \, \text{cm} \), we have:
\[ x^2 + y^2 = 10^2 = 100 \quad \text{(Equation 2)} \]

3. **Express \( y \) in terms of \( x \)**:
From Equation 1, we can express \( y \) in terms of \( x \):
\[ y = \frac{48}{x} \]

4. **Substitute \( y \) into Equation 2**:
Substitute \( y \) into Equation 2:
\[ x^2 + \left(\frac{48}{x}\right)^2 = 100 \]
Simplifying this gives:
\[ x^2 + \frac{2304}{x^2} = 100 \]

5. **Multiply through by \( x^2 \)** to eliminate the fraction:
\[ x^4 - 100x^2 + 2304 = 0 \]
Let \( z = x^2 \). Then we have:
\[ z^2 - 100z + 2304 = 0 \]

6. **Use the quadratic formula**:
The quadratic formula is given by:
\[ z = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Here, \( a = 1, b = -100, c = 2304 \):
\[ z = \frac{100 \pm \sqrt{(-100)^2 - 4 \times 1 \times 2304}}{2 \times 1} \]
\[ z = \frac{100 \pm \sqrt{10000 - 9216}}{2} \]
\[ z = \frac{100 \pm \sqrt{784}}{2} \]
\[ z = \frac{100 \pm 28}{2} \]
This gives us two potential values for \( z \):
\[ z = \frac{128}{2} = 64 \quad \text{or} \quad z = \frac{72}{2} = 36 \]

7. **Find \( x \)**:
Since \( z = x^2 \):
- If \( z = 64 \), then \( x = 8 \).
- If \( z = 36 \), then \( x = 6 \).

8. **Determine the shorter leg**:
The shorter leg \( x \) is \( 6 \, \text{cm} \) (since \( 8 \, \text{cm} \) would correspond to the longer leg \( y \)).

### Final Answer:
The length of the shorter leg is \( 6 \, \text{cm} \).