If the hypotenuse of a right-angled triangle is $41\text{ cm}$ and the area of the triangle is $180\text{ sq. cm}$, then the difference between the lengths of the legs of the triangle must be

### Concept & Algebraic Identities in Geometry We can determine properties of the legs of a right triangle without solving for them individually by utilizing algebraic identities. Let the legs be $x$ and $y$. From the Pythagorean theorem and area formula: $$ x^2 + y^2 = h^2 $$ $$ xy = 2 \times \text{Area} $$ We can find the difference $(x - y)$ using the identity: $(x - y)^2 = (x^2 + y^2) - 2xy$. ### Step-by-Step Solution 1. Let the lengths of the legs be $x$ and $y$. 2. We are given hypotenuse $h = 41$. Thus, $x^2 + y^2 = 41^2 = 1681$. 3. We are given Area $= 180$. Thus, $\frac{1}{2}xy = 180 \implies xy = 360$. 4. We need to find $|x - y|$. Use the algebraic identity: $(x - y)^2 = x^2 + y^2 - 2xy$ 5. Substitute the known values: $(x - y)^2 = 1681 - 2(360)$ $(x - y)^2 = 1681 - 720 = 961$ 6. Take the square root of both sides: $|x - y| = \sqrt{961} = 31\text{ cm}$. ### Exam Strategy & Shortcut Recognizing $41^2 = 1681$ quickly is crucial. If you are familiar with standard Pythagorean triplets, you might know $9, 40, 41$. Let's verify: $\frac{1}{2} \times 9 \times 40 = 180$, which matches the given area perfectly. The difference is $40 - 9 = 31$. This skip-steps the algebra entirely. ### Common Pitfall A common mistake is trying to solve the non-linear system of equations directly via substitution (e.g., $y = \frac{360}{x}$), which creates a complex quadratic equation in $x^2$ that consumes too much time. ### Final Answer Therefore, the correct answer is **$31\text{ cm}$**.