The perimeter of a right-angled triangle is $60\text{ cm}$. Its hypotenuse is $26\text{ cm}$. The area of the triangle is

### Concept & Algebraic Formulation You can deduce the area of a right triangle from its perimeter and hypotenuse without knowing the individual legs by using the identity: $$ (x + y)^2 = x^2 + y^2 + 2xy $$ Where $x$ and $y$ are the legs. The area is $\frac{1}{2}xy$. ### Step-by-Step Solution 1. Let the perpendicular legs be $x$ and $y$. 2. The perimeter is given as $60\text{ cm}$, and the hypotenuse is $26\text{ cm}$. $x + y + 26 = 60 \implies x + y = 34$. 3. By the Pythagorean theorem: $x^2 + y^2 = 26^2 = 676$. 4. Square the sum equation from step 2: $(x + y)^2 = 34^2$ $x^2 + y^2 + 2xy = 1156$ 5. Substitute the sum of squares from step 3: $676 + 2xy = 1156$ $2xy = 1156 - 676 = 480$ $xy = 240$. 6. Calculate the area: $\text{Area} = \frac{1}{2}xy = \frac{1}{2}(240) = 120\text{ cm}^2$. ### Exam Strategy & Shortcut If you observe the hypotenuse is 26, which is $13 \times 2$, you might recall the $5-12-13$ Pythagorean triplet. Scale it by 2 to get $10-24-26$. Check the perimeter: $10 + 24 + 26 = 60$. It matches! Calculate area directly: $\frac{1}{2} \times 10 \times 24 = 120$. ### Common Pitfall Avoid the trap of substituting $y = 34 - x$ into the Pythagorean theorem, as solving the resulting quadratic equation $(x^2 + (34-x)^2 = 676)$ is tedious and prone to arithmetic errors compared to utilizing algebraic identities. ### Final Answer Therefore, the correct answer is **$120\text{ cm}^2$**.