The hypotenuse of a right-angled isosceles triangle is $5\text{ cm}$. The area of the triangle is

### Concept & Right-Angled Isosceles Triangle A right-angled isosceles triangle has two equal perpendicular legs (let's call them $a$). By the Pythagorean theorem, the hypotenuse $h$ is given by $h = a\sqrt{2}$. The area of this triangle is given by: $$ \text{Area} = \frac{1}{2} \times a \times a = \frac{1}{2}a^2 $$ Alternatively, area can be directly related to the hypotenuse: $$ \text{Area} = \frac{h^2}{4} $$ ### Step-by-Step Solution 1. Let the equal legs of the right-angled isosceles triangle be $a$. 2. The given hypotenuse is $5\text{ cm}$. 3. Using the relationship $h = a\sqrt{2}$, we get $5 = a\sqrt{2}$, so $a = \frac{5}{\sqrt{2}}$. 4. Substitute $a$ into the area formula: $\text{Area} = \frac{1}{2} \times \left(\frac{5}{\sqrt{2}}\right)^2$ $\text{Area} = \frac{1}{2} \times \frac{25}{2}$ $\text{Area} = \frac{25}{4} = 6.25\text{ cm}^2$. ### Exam Strategy & Shortcut For a right-angled isosceles triangle, memorize the direct shortcut formula relating area and hypotenuse: $\text{Area} = \frac{h^2}{4}$. Squaring the hypotenuse gives $25$, and dividing by $4$ gives $6.25$ instantly. ### Common Pitfall Forgetting to square the $\sqrt{2}$ in the denominator when calculating the area is a common arithmetic error, leading to $\frac{25}{2} = 12.5$. ### Final Answer Therefore, the correct answer is **$6.25\text{ cm}^2$**.