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Geometry Question Please Help

by sanaa.rizwan » Tue Apr 09, 2013 6:42 pm
QR PS 157

A certain right angle triangle has length x, y & z. where x<y<z. if the area of this triangle region is 1. Which of the following indicates all of the possible values of y?

a. y > √ 2
b. √ 3 < y < √ 2
c. √ 2/3 < y < √ 3/2
d. √ 3/4 < y < √ 2/3
e. Y < √ 3/4
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by Brent@GMATPrepNow » Tue Apr 09, 2013 7:19 pm
sanaa.rizwan wrote:QR PS 157

A certain right angle triangle has length x, y & z. where x<y<z. if the area of this triangle region is 1. Which of the following indicates all of the possible values of y?

a. y > √ 2
b. √ 3 < y < √ 2
c. √ 2/3 < y < √ 3/2
d. √ 3/4 < y < √ 2/3
e. Y < √ 3/4
Since the triangle is a right triangle, the longest side (with length z) must be the hypotenuse.
This means the other two sides have lengths x and y.

IMPORTANT: these two sides form the right triangle, so let's call one side the base of the triangle and the other side the height.
Since the area is 1, we can take the area formula [area = (base)(height)/2] and plug in the lengths to get xy/2 = 1, which we can simplify to get xy = 2

Let's examine a situation in which x = y. In other words, the two legs of the right triangle have the same length.

So, we can take the equation, xy = 2 and replace x with y to get (y)(y) = 2
In other words, y^2 = 2
When we solve this equation for y, we get y = √2
So, if the two legs of the right triangle have the same length (in other words x = y), they would both have length equal to √2.

However, the question tells us that x < y

So, from this, we can conclude that y must be greater than √2

In other words, [spoiler]y > √2[/spoiler]

Answer = A

Cheers,
Brent
Brent Hanneson - Creator of GMATPrepNow.com
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