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Triangle XYZ is an isosceles right triangle. If side XY is l

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by BTGmoderatorDC » Fri Feb 08, 2019 6:13 pm

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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

OA C

Source: Veritas Prep
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by GMATGuruNY » Fri Feb 08, 2019 7:32 pm
BTGmoderatorDC 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
The sides of an isosceles right triangle are in the following ratio:
x - x - x√2, where x and x constitute the base and height and x√2 is the hypotenuse.

Since the triangle has a base of x, a height of x, and an area of 16, we get:
(1/2)(x)(x) = 16
x² = 32
x = √32

Since XY is longer than YZ, XY must be the hypotenuse and thus have a length of x√2:
x√2 = √32 * √2 = √64 = 8

The correct answer is C.
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by Brent@GMATPrepNow » Sat Feb 09, 2019 2:11 pm
BTGmoderatorDC 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
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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by Scott@TargetTestPrep » Fri Feb 15, 2019 5:14 pm
BTGmoderatorDC 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

OA C

Source: Veritas Prep
Recall that an isosceles right triangle is a 45-45-90 triangle, which has two equal legs and a ratio of x : x : x√2 for its three sides. Since XY is longer than YZ, XY must be the hypotenuse. If we let x = the length of leg YZ, we can use the formula for the area of a triangle, ½ bh, to create the equation:

x^2/2 = 16

x^2 = 32

x = √32 = √16 x √2 = 4√2

So leg YZ = 4√2, and thus, hypotenuse XY is 4√2 x √2 = 8.

Answer: C

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