An isosceles right triangle has area `8cm^(2)` .Find the length of hypotenuse.

Options

A. `sqrt(32)cm`
B. `sqrt(16)cm`
C. `sqrt(48)cm`
D. `sqrt(24)cm`

Answer

To find the length of the hypotenuse of an isosceles right triangle with an area of \(8 \, \text{cm}^2\), we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Triangle Properties**:
- In an isosceles right triangle, the two legs (let's call them \(a\)) are equal, and the area can be calculated using the formula:
\[ \text{Area} = \frac{1}{2} \times \text{base} \times \text{height} \]
- Since the base and height are the same in this case, we can say:
\[ \text{Area} = \frac{1}{2} \times a \times a = \frac{1}{2} a^2 \]

2. **Set Up the Equation**:
- Given that the area is \(8 \, \text{cm}^2\), we can set up the equation:
\[ \frac{1}{2} a^2 = 8 \]

3. **Solve for \(a^2\)**:
- Multiply both sides by \(2\) to eliminate the fraction:
\[ a^2 = 16 \]

4. **Find the Length of the Legs**:
- Take the square root of both sides to find \(a\):
\[ a = \sqrt{16} = 4 \, \text{cm} \]

5. **Use the Pythagorean Theorem to Find the Hypotenuse**:
- In a right triangle, the hypotenuse \(c\) can be found using the Pythagorean theorem:
\[ c^2 = a^2 + a^2 \]
- Substitute \(a = 4 \, \text{cm}\):
\[ c^2 = 4^2 + 4^2 = 16 + 16 = 32 \]

6. **Calculate the Length of the Hypotenuse**:
- Take the square root of both sides:
\[ c = \sqrt{32} = 4\sqrt{2} \, \text{cm} \]

### Final Answer:
The length of the hypotenuse is \(4\sqrt{2} \, \text{cm}\).

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