An isoceles right angle triangle has area 200 cm^2. Then length of its hypotenuse is:-

Options

A. a. 15√2
B. b. 10/√2
C. c. 10√2
D. d. 20√2

Answer

To solve the problem of finding the length of the hypotenuse of an isosceles right triangle with an area of 200 cm², we can follow these steps:

### Step-by-Step Solution:

1. **Understand the properties of the triangle**:
An isosceles right triangle has two sides equal and one right angle (90 degrees). Let's denote the equal sides as \( b \).

2. **Use the area formula for triangles**:
The area \( A \) of a triangle can be calculated using the formula:
\[ A = \frac{1}{2} \times \text{Base} \times \text{Height} \]
For our isosceles right triangle, both the base and height are equal to \( b \). Therefore, the area can be expressed as:
\[ A = \frac{1}{2} \times b \times b = \frac{1}{2} b^2 \]

3. **Set up the equation with the given area**:
We know the area is 200 cm², so we can set up the equation:
\[ \frac{1}{2} b^2 = 200 \]

4. **Solve for \( b^2 \)**:
Multiply both sides by 2 to eliminate the fraction:
\[ b^2 = 200 \times 2 = 400 \]

5. **Find \( b \)**:
Take the square root of both sides to find \( b \):
\[ b = \sqrt{400} = 20 \text{ cm} \]

6. **Use the Pythagorean theorem to find the hypotenuse**:
The hypotenuse \( AC \) can be calculated using the Pythagorean theorem:
\[ AC^2 = AB^2 + BC^2 \]
Since \( AB = b \) and \( BC = b \):
\[ AC^2 = b^2 + b^2 = 2b^2 \]
Substitute \( b^2 = 400 \):
\[ AC^2 = 2 \times 400 = 800 \]

7. **Calculate \( AC \)**:
Take the square root to find the hypotenuse:
\[ AC = \sqrt{800} = \sqrt{400 \times 2} = \sqrt{400} \times \sqrt{2} = 20\sqrt{2} \text{ cm} \]

### Final Answer:
The length of the hypotenuse is \( 20\sqrt{2} \) cm.

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