Right triangle ABC has sides with length x, y and z. If triangle ABC

Question

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

* kudos for all correct solutions

EDIT: Changed perimeter to 17 (sorry)

BrentGMATPrepNow · Accepted Answer

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

rohit8865 · Answer

Sure!!!

Please follow below attached fig.

As x^2+y^2=z^2----(1)

Also given x^2+y^2+z^2=98
putting value z^2 from (1)

2(x^2+y^2)= 98
or x^2+y^2=49------(2)

again putting x^2+y^2 =z^2 from (1) 
z^2 =49 or z=7

Also given that perimeter x+y+z= 17
thus x+y= 17-7 =10---------(3) 
sq both side to get 
x^2+y^2+2xy =100
from (2) 
2xy =100-49 =51
Also area = xy/2= 51/4 = 12.75

Ans 12.75

Ans A

kavyagupta · Answer

please share the solution

bdh009 · Answer

Interesting question - how would you know to square both sides? Is there a rule?

LeonidK · Answer

x^2 + y^2 + z^2 = 98 
 z^2 = x^2 + y^2 
 2*z^2 = 98 
z = 7

given perimeter: x + y + z = 17
if z = 7, than x+y = 10

(x+y)^2 = x^2 + 2*x*y + y^2 
 x^2 + 2*x*y + y^2 = (x+y)^2 = 10^2 = 100 
 x^2 + y^2 = 49 
 2*x*y = 51 
 x*y = 25.5

Area of triangle = 1/2*x*y = 12.75

mdsaddamforgmat · Answer

since, there are no specific mentions of the placement of angle A,B& C and the sides... is it always ok to consider that "B" will be 90deg and "z" will be hypotenuse ?

rohit8865 · Answer

dear mdsaddamforgmat the answer to your question is NO. here as we calculated z=7 ,it means it is the longest side of rt. triangle ie hypotenuse. try making other side the longest one,...u will not get a rt triangle then. hope it helps.

VIVA1060 · Answer

Hi Members,

After I get the hypotenuse = 7 (diameter of the circle). When I compute the area as 1/2*7*3.5 (radius) = 12.25

I don't get the answer as 12.75. Can you help me understand where I am going wrong?

bumpbot · Answer

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Right triangle ABC has sides with length x, y and z. If triangle ABC

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