The area of a right angled isosceles triangle whose hypotenuse is equal to 270 m is

Options

A. `19000 m^(2)`
B. `18225 m^(2)`
C. `17256 m^(2)`
D. `18325 m^(2)`

Answer

To find the area of a right-angled isosceles triangle with a hypotenuse of 270 m, we can follow these steps:

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

### Step 2: Apply the Pythagorean theorem
According to the Pythagorean theorem:
\[ \text{Hypotenuse}^2 = \text{Leg}_1^2 + \text{Leg}_2^2 \]
Since both legs are equal, we can write:
\[ \text{Hypotenuse}^2 = A^2 + A^2 = 2A^2 \]
Given that the hypotenuse is 270 m, we can substitute this value into the equation:
\[ 270^2 = 2A^2 \]

### Step 3: Calculate \( 270^2 \)
Calculating \( 270^2 \):
\[ 270^2 = 72900 \]
So, we have:
\[ 72900 = 2A^2 \]

### Step 4: Solve for \( A^2 \)
To find \( A^2 \), divide both sides by 2:
\[ A^2 = \frac{72900}{2} = 36450 \]

### Step 5: Calculate the area of the triangle
The area \( A \) of a right-angled triangle is given by:
\[ \text{Area} = \frac{1}{2} \times \text{Base} \times \text{Height} \]
In this case, both the base and height are equal to \( A \):
\[ \text{Area} = \frac{1}{2} \times A \times A = \frac{1}{2} \times A^2 \]
Substituting the value of \( A^2 \):
\[ \text{Area} = \frac{1}{2} \times 36450 = 18225 \text{ m}^2 \]

### Final Answer
The area of the right-angled isosceles triangle is \( 18225 \text{ m}^2 \).

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