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

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:


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:


This also means the last remaining vertex must be Z:


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


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.


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

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