The base of a prism is a right angled triangle, the length of whose hypotenuse is 10 cm. If the lateral surface area of the prism be 384 `cm^ 2` and its height be 16 cm. The other two sides of its base is

Options

A. 8 cm, 6 cm
B. 12 cm, 14 cm
C. 12 cm, 12 cm
D. None of the above

Answer

To solve the problem, we need to find the lengths of the other two sides of a right-angled triangle whose hypotenuse is given as 10 cm, and the lateral surface area of the prism formed by this triangle is 384 cm² with a height of 16 cm.

### Step-by-Step Solution:

1. **Identify the Given Information:**
- Hypotenuse (c) = 10 cm
- Lateral Surface Area (LSA) = 384 cm²
- Height (h) = 16 cm

2. **Use the Formula for Lateral Surface Area of a Prism:**
The formula for the lateral surface area of a prism is given by:
\[ \text{LSA} = \text{Perimeter of base} \times \text{Height} \]
Let the other two sides of the right-angled triangle be \( x \) and \( y \). The perimeter of the triangle is:
\[ \text{Perimeter} = x + y + 10 \]
Therefore, we have:
\[ 384 = (x + y + 10) \times 16 \]

3. **Simplify the Equation:**
Dividing both sides by 16:
\[ x + y + 10 = \frac{384}{16} = 24 \]
Rearranging gives:
\[ x + y = 24 - 10 = 14 \quad \text{(Equation 1)} \]

4. **Apply the Pythagorean Theorem:**
For a right-angled triangle, the Pythagorean theorem states:
\[ x^2 + y^2 = c^2 \]
Substituting \( c = 10 \):
\[ x^2 + y^2 = 10^2 = 100 \quad \text{(Equation 2)} \]

5. **Substitute Equation 1 into Equation 2:**
From Equation 1, we can express \( y \) in terms of \( x \):
\[ y = 14 - x \]
Substitute \( y \) into Equation 2:
\[ x^2 + (14 - x)^2 = 100 \]

6. **Expand and Simplify:**
Expanding \( (14 - x)^2 \):
\[ x^2 + (196 - 28x + x^2) = 100 \]
Combine like terms:
\[ 2x^2 - 28x + 196 = 100 \]
Rearranging gives:
\[ 2x^2 - 28x + 96 = 0 \]
Dividing the entire equation by 2:
\[ x^2 - 14x + 48 = 0 \]

7. **Factor the Quadratic Equation:**
We need to factor the equation:
\[ (x - 8)(x - 6) = 0 \]
Thus, the solutions for \( x \) are:
\[ x = 8 \quad \text{or} \quad x = 6 \]

8. **Find the Corresponding Values of \( y \):**
If \( x = 8 \):
\[ y = 14 - 8 = 6 \]
If \( x = 6 \):
\[ y = 14 - 6 = 8 \]

9. **Conclusion:**
The lengths of the other two sides of the triangle are:
\[ 6 \, \text{cm and} \, 8 \, \text{cm} \]

### Final Answer:
The lengths of the other two sides of the triangle are **6 cm and 8 cm**.