If the perimeter of the isosceles right triangle shown is 1 + √2, what

In this question on isosceles right angled triangles, taking a logical approach will help you solve the problem faster than taking a more conventional approach of rationalizing surds.

Let us see how we can do it, with the help of this diagram:


As you can observe from the solution shown in the diagram, it’s very natural to equate the rational and the irrational parts to each other. But, in doing so, you see that there are conflicting values for x. What happens at this stage is some of us may get stuck and wonder where we are wrong!

All that we need to realise is that it can be the other way round also – the equal sides can add up to give us √2 and the hypotenuse can be 1. Here, we see that the value of x is consistent when we equate the rational and the irrational sides respectively. This proves conclusively that it is this case that is being stated in the question and not the other way round.

The perpendicular sides are \(\frac{1}{√2}\) each. The area of a right angled triangle = ½ * Product of the perpendicular sides.
Therefore, area of the given isosceles right triangle = ½ * \(\frac{1}{√2}\) * \(\frac{1}{√2}\) = ¼.

The correct answer option is A.

Hope that helps!