The area of an isosceles right - angled triangle is 72 `cm^(2)` .Find its hypotenuse .

Answer

To find the hypotenuse of an isosceles right-angled triangle with an area of 72 cm², we can follow these steps:

### Step 1: Understand the properties of the triangle
In an isosceles right-angled triangle, the two legs (the sides that form the right angle) are equal in length. Let's denote the length of each leg as \( x \).

### Step 2: Write the formula for the area of a triangle
The area \( A \) of a triangle is given by the formula:
\[ A = \frac{1}{2} \times \text{base} \times \text{height} \]
For our isosceles right-angled triangle, both the base and height are equal to \( x \). Therefore, the area can be expressed as:
\[ A = \frac{1}{2} \times x \times x = \frac{1}{2} x^2 \]

### Step 3: Set up the equation using the given area
We know the area of the triangle is 72 cm², so we can set up the equation:
\[ \frac{1}{2} x^2 = 72 \]

### Step 4: Solve for \( x^2 \)
To eliminate the fraction, multiply both sides of the equation by 2:
\[ x^2 = 72 \times 2 \]
\[ x^2 = 144 \]

### Step 5: Solve for \( x \)
Now, take the square root of both sides to find \( x \):
\[ x = \sqrt{144} = 12 \text{ cm} \]

### Step 6: Use the Pythagorean theorem to find the hypotenuse
In an isosceles right-angled triangle, the hypotenuse \( c \) can be found using the Pythagorean theorem:
\[ c^2 = x^2 + x^2 = 2x^2 \]
Substituting \( x = 12 \) cm:
\[ c^2 = 2(12^2) = 2(144) = 288 \]
Now, take the square root to find \( c \):
\[ c = \sqrt{288} = \sqrt{144 \times 2} = 12\sqrt{2} \text{ cm} \]

### Final Answer
The hypotenuse of the triangle is \( 12\sqrt{2} \) cm.

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