The diagonal of a right angle isosceles triangle is 5 cm. Its area will be

Options

A. 5 sq. cm
B. 6.25 sq. cm
C. 6.50 sq. cm.
D. 12.5 sq. cm.

Answer

To find the area of a right angle isosceles triangle with a diagonal of 5 cm, we can follow these steps:

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

### Step 2: Apply the Pythagorean theorem
According to the Pythagorean theorem:
\[ \text{(Leg)}^2 + \text{(Leg)}^2 = \text{(Hypotenuse)}^2 \]
In our case, the hypotenuse (the diagonal) is 5 cm. Therefore:
\[ x^2 + x^2 = 5^2 \]
This simplifies to:
\[ 2x^2 = 25 \]

### Step 3: Solve for \( x^2 \)
Dividing both sides by 2 gives:
\[ x^2 = \frac{25}{2} \]

### Step 4: Find \( x \)
Taking the square root of both sides, we find:
\[ x = \sqrt{\frac{25}{2}} = \frac{5}{\sqrt{2}} \text{ cm} \]

### Step 5: Calculate the area of the triangle
The area \( A \) of a triangle is given by the formula:
\[ A = \frac{1}{2} \times \text{base} \times \text{height} \]
In this case, both the base and height are equal to \( x \):
\[ A = \frac{1}{2} \times x \times x = \frac{1}{2} \times \left(\frac{5}{\sqrt{2}}\right) \times \left(\frac{5}{\sqrt{2}}\right) \]
This simplifies to:
\[ A = \frac{1}{2} \times \frac{25}{2} = \frac{25}{4} \text{ cm}^2 \]

### Step 6: Final answer
Converting \( \frac{25}{4} \) to a decimal gives:
\[ A = 6.25 \text{ cm}^2 \]

Thus, the area of the triangle is **6.25 cm²**.

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