An Isosceles Right Triangle has an Area of 50. What is the Length of the Hypotenuse? GMAT Problem Solving

Question: An isosceles right triangle has an area of 50. What is the length of the hypotenuse?

1. 5
2. \(5\sqrt2\)
3. \(5\sqrt3\)
4. 10
5. \(10\sqrt2\)

‘An isosceles right triangle has an area of 50. What is the length of the hypotenuse?’ - is a topic of the GMAT Quantitative reasoning section of GMAT. This question has been taken from the book “Official Guide for GMAT Reviews”. To solve GMAT Problem Solving questions a student must have knowledge about a good amount of qualitative skills. The GMAT Quant topic in the problem-solving part requires calculative mathematical problems that should be solved with proper mathematical knowledge.

Solution and Explanation:

Approach Solution 1:

As it is given in the question that it is an isosceles triangle. Then let the sides be x

Area of isosceles triangle = \(\frac{1}{2} * x* x = 50\)

implies that \(x^2\) = 100

implies that x = 100

So the hypotenuse of the isosceles triangle = \(x\sqrt2=10\sqrt2\)

Therefore the option E is the correct answer.

Correct Answer: E

Approach Solution 2:

Here's an isosceles right triangle with sides of length x

Area of triangle = (base)(height)/2
So, we can write: 50 = (x)(x)/2
implies that 50 = x²/2
implies that 100 = x²
implies that x = 10

So, an isosceles triangle looks like this:

To find the length of the hypotenuse (k), we can apply the Pythagorean Theorem:
So, from the problem we get: 10² + 10² = k²
Further equating 200 = k²

So, k =\(\sqrt{200}\)

= \(\sqrt{(100)(2)}\)

= (\(\sqrt{100}\))(\(\sqrt{2}\))

= 10\(\sqrt{2}\)

Therefore the option E is the correct answer.

Correct Answer: E

Approach Solution 3:

Let us consider the sides of isosceles triangle be x
then the Area becomes (1/2)∗x∗x=50=(1/2)∗x∗x=50
that implies x2=100x2=100
that implies x=10x=10
Hypotenuse becomes x√2=10*√2=10√2

Correct Answer: E

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