Question

If the perimeter of an isosceles right angled triangle is (12 + 6"12) m, then what is the area of the triangle?

• A. 9 m²
• B. 18 m²
• C. 36 m²
• D. 81 m²

Answer 1

The Correct Option is B

Finding the Area of an Isosceles Right-Angled Triangle 

Step 1: Define the Perimeter of the Triangle

Let the legs of the isosceles right-angled triangle each have length \( x \). The hypotenuse is:

\[ x\sqrt{2} \]

Therefore, the perimeter is:

\[ x + x + x\sqrt{2} = 2x + x\sqrt{2} \]

Step 2: Solve for \( x \)

We are given that the perimeter is:

\[ 12 + 6\sqrt{2} \]

Setting up the equation:

\[ 2x + x\sqrt{2} = 12 + 6\sqrt{2} \]

Factor out \( x \):

\[ x(2 + \sqrt{2}) = 12 + 6\sqrt{2} \]

Observing that \( x = 6 \) satisfies the equation:

\[ 6(2 + \sqrt{2}) = 12 + 6\sqrt{2} \]

Thus, the length of each leg is \( 6 \) meters.

Step 3: Calculate the Area

The area of a right-angled triangle is given by:

\[ \text{Area} = \frac{1}{2} \times \text{leg}_1 \times \text{leg}_2 \]

Substituting the values:

\[ \text{Area} = \frac{1}{2} \times 6 \times 6 \]

\[ = \frac{36}{2} = 18 \text{ m}^2 \]

Final Answer:

Thus, the area of the triangle is 18 m².