An isosceles right-angled triangle has area 200 cmWhat is the length of its hypotenuse?

Answer

To find the length of the hypotenuse of an isosceles right-angled triangle with an area of 200 cm², we can follow these steps:

### Step 1: Understand the properties of the triangle
In an isosceles right-angled triangle, the two legs (let's denote them as \( a \)) are equal in length, and the area can be calculated using the formula:
\[ \text{Area} = \frac{1}{2} \times \text{base} \times \text{height} \]
Since both the base and height are equal to \( a \), we can rewrite the area formula as:
\[ \text{Area} = \frac{1}{2} \times a \times a = \frac{1}{2} a^2 \]

### Step 2: Set up the equation for the area
Given that the area is 200 cm², we can set up the equation:
\[ \frac{1}{2} a^2 = 200 \]

### Step 3: Solve for \( a^2 \)
To eliminate the fraction, multiply both sides of the equation by 2:
\[ a^2 = 400 \]

### Step 4: Solve for \( a \)
Now, take the square root of both sides to find \( a \):
\[ a = \sqrt{400} = 20 \text{ cm} \]

### Step 5: Use the Pythagorean theorem to find the hypotenuse
In a right-angled triangle, the hypotenuse \( c \) can be found using the Pythagorean theorem:
\[ c^2 = a^2 + a^2 \]
Substituting \( a = 20 \):
\[ c^2 = 20^2 + 20^2 = 400 + 400 = 800 \]

### Step 6: Solve for \( c \)
Now, take the square root of both sides to find the hypotenuse:
\[ c = \sqrt{800} \]

### Step 7: Simplify \( \sqrt{800} \)
To simplify \( \sqrt{800} \):
\[ \sqrt{800} = \sqrt{400 \times 2} = \sqrt{400} \times \sqrt{2} = 20\sqrt{2} \]

### Conclusion
Thus, the length of the hypotenuse is:
\[ \text{Hypotenuse} = 20\sqrt{2} \text{ cm} \]

---