Problem Solving

The Perimeter of a certain isosceles right triangle is 16 + 16√2. What is the length of the hypotenuse of the triangle?

A) 8
B) 16
C) 4√2
D) 8√2
E) 16√2

An IMPORTANT point to remember is that, in any isosceles right triangle, the sides have length x, x, and x√2 for some positive value of x.

Note: x√2 is the length of the hypotenuse, so our goal is to find the value of x√2

From here, we can see that the perimeter will be x + x + x√2

In the question, the perimeter is 16 + 16√2, so we can create the following equation:
x + x + x√2 = 16 + 16√2,
Simplify: 2x + x√2 = 16 + 16√2
IMPORTANT: Factor x√2 from the left side to get : x√2(√2 + 1) = 16 + 16√2
Now factor 16 from the right side to get: x√2(√2 + 1) = 16(1 + √2)
Divide both sides by (1 + √2) to get: x√2 = 16

Answer = B

Cheers,
Brent

Hi bml1105,

Mitch's approach to TEST THE ANSWERS is perfect for this scenario. There's a Number Property about Isosceles triangles that will help you to avoid most of the wrong answers...

The ratio of sides in an Isosceles Triangle is: x, x, and x√2

If, for example, one of the "short sides" was 5, then we'd have 5, 5, 5√2 and the perimeter would be 10 + 5√2. Notice that the 10 and 5 are different numbers....That will ALWAYS happen IF the short side is an integer and the hypotenuse has a √2 attached to it....

We're given a perimeter of 16 + 16√2; here, both numbers are 16, so we are NOT dealing with the above scenario. The "short sides" CANNOT be integers and the hypotenuse CANNOT have a √2 attached to it. Eliminate C, D and E.

From there, we can TEST A and B to find the match.

GMAT assassins aren't born, they're made,
Rich

bml1105 wrote:
Hi Brent,

I'm a little confused. How can we factor out x√2 from 2x + x√2 and get x√2(√2 + 1) when the 2x doesn't have an x√2. If anything wouldn't it be 2x/x√2 (√2/√2) = (2√2)/2?

Good question.
Notice that 2 = (√2)(√2)
So, let's start at the point where we have: 2x + x√2 = 16 + 16√2
We can rewrite this as (√2)(√2)x + x√2 = 16 + 16√2
So, on the left side, both terms have an x and a √2
So we can factor x√2 from the left side to get : x√2(√2 + 1) = 16 + 16√2

I hope that helps.

Cheers,
Brent

Don't be too hard on yourself, Garmahis. This is a tricky question. In fact, I'd put it in the 700 range (difficulty-wise).

Cheers,
Brent