Tricky - Right triangle ABC has sides with length x, y and z

[Brent@GMATPrepNow]

Right triangle ABC has sides with length x, y and z. If triangle ABC has perimeter 17, and x² + y² + z² = 98, then what is the area of triangle ABC?

A) 12.75
B) 13.25
C) 14
D) 14.5
E) 15.25

Answer: A

Source: www.gmatprepnow.com
Difficulty level: 700+

[ByjusGMATapp]

let x^2 + y^2 = z^2
x^2 +y^2 +z^2 = 98

substituting, we get
2. z^2 = 98
or z = 7
the hypotenuse is 7
therefore
x +y +7 = 17
or x +y = 10
and x^2 + y^2 = 49
or (x + y)^2 - 2xy = 49

replacing x + y =10, we get
10^2 - 2xy = 49
or 2xy = 51
ir xy/2 = 51/4
or area = 12.75

option A

[DavidG@VeritasPrep]

A useful reminder: anytime you need to solve for some variation of a*b, and you're given one equation that defines a + b and a second that defines a² + b², you can create that a*b term by squaring both sides of the a + b equation. Here's a good example of a question where this can come in handy, should you approach algebraically: https://www.beatthegmat.com/ps-diagonal- ... 81412.html

[Brent@GMATPrepNow]

Right triangle ABC has sides with length x, y and z. If triangle ABC has perimeter 17, and x² + y² + z² = 98, then what is the area of triangle ABC?

A) 12.75
B) 13.25
C) 14
D) 14.5
E) 15.25

Answer: A

Source: www.gmatprepnow.com
Difficulty level: 700+

Let z = length of hypotenuse
Let x and y = lengths of the sides that make up the triangle's right angle
NOTE: since triangle area = (base)(height)/2, we know that the area of triangle ABC = xy/2

Given: x² + y² + z² = 98
The Pythagorean Theorem tells us that x² + y² = z²
So, we can write: x² + y² + (x² + y²) = 98
Rewrite: 2(x² + y²) = 98
Or we can write: x² + y² = 49

Since we already know that x² + y² = z², we can conclude that z² = 49, which means z = 7

If the perimeter = 17, we can write: x + y + z = 17
Since z = 7, we can write: x + y + 7 = 17
Simplify to get: x + y = 10

IMPORTANT: If we square both sides of the above equation, some nice things happen.
(x + y) ² = 10²
Expand: x² + 2xy + y² = 100
Rearrange to get: (x² + y²) + 2xy = 100
Replace x² + y² with 49 to get: 49 + 2xy = 100
Simplify: 2xy = 51

NOTE: Our goal is to determine the value of xy/2
So, take 2xy = 51 and divide both sides by 4 to get: xy/2 = 51/4 = 12.75

Answer: A