An isosceles right triangle has area 8 `cm^(2)`. The length of its hypotenuse is

Options

A. `sqrt32` cm
B. `sqrt16` cm
C. `sqrt48` cm
D. `sqrt24`

Explanation

To find the length of the hypotenuse of an isosceles right triangle with an area of 8 cm², we can follow these steps:

### Step 1: Understand the properties of an isosceles right triangle.
In an isosceles right triangle, the two legs (let's call them \( a \)) are equal, and the area can be calculated using the formula:
\[ \text{Area} = \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times a \times a = \frac{1}{2} a^2 \]

### Step 2: Set up the equation for the area.
Given that the area is 8 cm², we can set up the equation:
\[ \frac{1}{2} a^2 = 8 \]

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

### Step 4: Solve for \( a \).
Taking the square root of both sides gives:
\[ a = \sqrt{16} = 4 \, \text{cm} \]

### Step 5: Calculate the hypotenuse.
In an isosceles right triangle, the hypotenuse \( c \) can be found using the Pythagorean theorem:
\[ c = \sqrt{a^2 + a^2} = \sqrt{2a^2} = a\sqrt{2} \]
Substituting \( a = 4 \):
\[ c = 4\sqrt{2} \, \text{cm} \]

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

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