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Geometry

by Obinna.uwadoka » Thu Nov 21, 2013 1:54 pm
If the perimeter of an isosceles right triangle is 16+16root 2, what is the length of the hypotenuse. ( the root should be square root sign, but I can't find the sign on my iPad). Someone please help with the answer to the question. Thanks
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by GMATGuruNY » Thu Nov 21, 2013 2:39 pm
When you post a problem, please include the answer choices.
The perimeter of a certain isosceles right triangle is 16 + 16√2. What is the length of the hypotenuse?
a. 8
b. 16
c. 4√2
d. 8√2
e. 16√2
The sides of an isosceles right triangle are proportioned: s : s : s√2.
So if s=side and h=hypotenuse, then h = s√2 and s = h/√2.

We can plug in the answers, which represent the hypotenuse.

Answer choice C: h = 4√2
s = (4√2)/√2 = 4.
p = 4 + 4 + 4√2 = 8 + 4√2.
Eliminate C. The perimeter needs to be quite a bit larger.

Answer choice B: h = 16
s = 16/√2 = (16*√2)/(√2*√2) = (16√2)/2 = 8√2.
p = 8√2 + 8√2 + 16 = 16 + 16√2. Success!

The correct answer is B.
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by Brent@GMATPrepNow » Thu Nov 21, 2013 2:54 pm
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
Here's another approach:

An important point here 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

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by [email protected] » Thu Nov 21, 2013 10:29 pm
Hi Obinna.uwadoka,

Both Mitch and Brent have provided solid explanations to this question. It's worth noting that Geometry questions always involve very specific math formulas and patterns.

By definition, an isosceles right triangle is a 45/45/90 triangle. This type of question comes down to your knowledge of the ratio of the sides: X/X/Xroot2

This means that the third side will be the product of one of the short sides and (root2); whether you actually SEE a (root2) or not is optional.

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