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Re: Triangle XYZ is an isosceles right triangle. If side XY is longer than [#permalink]
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Carcass wrote:
Triangle XYZ is an isosceles right triangle. If side XY is longer than side YZ, and the area of the triangle is 16, what is the measure of side XY?

A. 4
B. 4√2
C. 8
D. 8√2
E. Cannot be determined from the information provided

Kudos for the right answer and explanation
Question part of the project GRE Quantitative Reasoning Daily Challenge - (2021) EDITION
GRE - Math Book


Triangle XYZ is an isosceles right triangle.
Let's sketch an isosceles right triangle:
Image

Side XY is longer than side YZ
Since the hypotenuse is the longest side of a right triangle, side XY must be the hypotenuse. Add this to our diagram:
Image

This also means the last remaining vertex must be Z:
Image

The area of the triangle is 16. What is the measure of side XY?
Let j = the length of side ZY,
Since ZY = ZX, we can see that side ZX must also halve length j
Image

Area of triangle = (base)(height)/2
So, we can write: 16 = (j)(j)/2
Simplify: 16 = j²/2
Multiply both sides by 2 to get: 32 = j²
Solve: j = √32

What is the measure of side XY?
Our diagram now looks like this.
Image

Applying the Pythagorean Theorem, we can write: (√32)² + (√32)² = c²
Simplify: 32 + 32 = c²
Simplify: 64 = c²
Solve: c = 8

Answer: C

Cheers,
Brent
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Re: Triangle XYZ is an isosceles right triangle. If side XY is longer than [#permalink]
1
Let's break down the given information:
1) The triangle is an isosceles triangle, which means 2 sides have equal length.
2) Side XY is greater than YZ -> this means that the side XY is the hypotenuse because in a right-angled triangle, the hypotenuse is always the biggest side and it also is an isosceles triangle. Hence, XY is the hypotenuse.
3) The area of the triangle is 16
X
|\
| \
|__\
Z ....Y

Area of triangle = base*height/2
16=ZY*XZ/2
as XZ=ZY we can write
16=XZ*XZ/2
16*2=XZ^2
XZ^2=32
XZ=root(32)
XZ=4root(2)

As it is a 45-45-90 triangle that means the sides are in ratio 1:1:root2

So 4root2 * root2 = 8

Answer is C
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Re: Triangle XYZ is an isosceles right triangle. If side XY is longer than [#permalink]
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