The perimeter of an isosceles right-angled triangle is 2 meters. Find the length of its longer side.

1. Analyze the question

We are given an isosceles right-angled triangle with a perimeter of 2 meters. We need to find the length of the "longer side," which in this case refers to the hypotenuse.

2. Explain the concept

• Isosceles Right-Angled Triangle: This triangle has two equal sides (let's call their length $x$) and one hypotenuse ($h$).
• Pythagoras Theorem: For a right-angled triangle, $a^2 + b^2 = c^2$. Here, $x^2 + x^2 = h^2$, which simplifies to $2x^2 = h^2$, so $h = x\sqrt{2}$.
• Perimeter: The sum of all sides is $x + x + h = 2$.

3. Show the solution

Step 1: Set up the perimeter equation. $2x + h = 2$ Since $h = x\sqrt{2}$, we substitute $h$ into the equation: $2x + x\sqrt{2} = 2$

Step 2: Solve for $x$ (the equal sides). Factor out $x$: $x(2 + \sqrt{2}) = 2$ $x = \frac{2}{2 + \sqrt{2}}$

Step 3: Rationalize the denominator. Multiply the numerator and denominator by the conjugate $(2 - \sqrt{2})$: $x = \frac{2(2 - \sqrt{2})}{(2 + \sqrt{2})(2 - \sqrt{2})}$ $x = \frac{4 - 2\sqrt{2}}{4 - 2}$ $x = \frac{4 - 2\sqrt{2}}{2}$ $x = 2 - \sqrt{2}$

Step 4: Find the hypotenuse ($h$). We know $h = x\sqrt{2}$. Substitute the value of $x$ we just found: $h = (2 - \sqrt{2}) \times \sqrt{2}$ $h = 2\sqrt{2} - (\sqrt{2} \times \sqrt{2})$ $h = 2\sqrt{2} - 2$ Rearranging this gives: $-2 + 2\sqrt{2}$

4. Identify the answer

Comparing our result with the options provided: A. $2 - \sqrt{2}$ B. $-4 + 3\sqrt{2}$ C. It cannot be determined D. $-2 + 2\sqrt{2}$ m

The calculated value matches option D.

5. Tip

When dealing with surds in geometry, always remember to rationalize the denominator early. Also, remember that in an isosceles right-angled triangle, the ratio of the sides is always $1 : 1 : \sqrt{2}$. If you know the perimeter, you can quickly find the sides by dividing the perimeter by $(2 + \sqrt{2})$.