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

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GMAT Club Legend
Joined: 11 Sep 2015
Posts: 4886
GMAT 1: 770 Q49 V46
Right triangle ABC has sides with length x, y and z. If triangle ABC  [#permalink]

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Updated on: 27 Mar 2017, 05:36
Top Contributor
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Difficulty:

65% (hard)

Question Stats:

64% (03:07) correct 36% (03:09) wrong based on 185 sessions

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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)

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Originally posted by BrentGMATPrepNow on 26 Mar 2017, 05:55.
Last edited by BrentGMATPrepNow on 27 Mar 2017, 05:36, edited 2 times in total.
GMAT Club Legend
Joined: 11 Sep 2015
Posts: 4886
GMAT 1: 770 Q49 V46
Re: Right triangle ABC has sides with length x, y and z. If triangle ABC  [#permalink]

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27 Mar 2017, 05:35
6
Top Contributor
7
GMATPrepNow wrote:
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

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 = [color=red]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

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Joined: 05 Mar 2015
Posts: 936
Right triangle ABC has sides with length x, y and z. If triangle ABC  [#permalink]

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Updated on: 27 Mar 2017, 09:29
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1
kavyagupta wrote:

Sure!!!

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
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Originally posted by rohit8865 on 26 Mar 2017, 18:16.
Last edited by rohit8865 on 27 Mar 2017, 09:29, edited 1 time in total.
##### General Discussion
Intern
Joined: 01 Dec 2016
Posts: 2
Re: Right triangle ABC has sides with length x, y and z. If triangle ABC  [#permalink]

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26 Mar 2017, 16:56
1
Current Student
Joined: 12 Jan 2017
Posts: 5
Location: United States
GMAT 1: 670 Q44 V38
GPA: 3.26
WE: Marketing (Consumer Products)
Re: Right triangle ABC has sides with length x, y and z. If triangle ABC  [#permalink]

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24 Apr 2017, 17:26
Interesting question - how would you know to square both sides? Is there a rule?
Manager
Joined: 28 Jul 2016
Posts: 128
Re: Right triangle ABC has sides with length x, y and z. If triangle ABC  [#permalink]

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19 Jun 2017, 10:25
henkbdh828 wrote:
Interesting question - how would you know to square both sides? Is there a rule?

$$x^2 + y^2 + z^2 = 98$$
$$z^2 = x^2 + y^2$$
$$2*z^2 = 98$$
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
Intern
Joined: 05 Dec 2019
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GMAT 1: 420 Q33 V15
GMAT 2: 640 Q49 V27
Right triangle ABC has sides with length x, y and z. If triangle ABC  [#permalink]

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21 Feb 2020, 19:06
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 ?
Director
Joined: 05 Mar 2015
Posts: 936
Re: Right triangle ABC has sides with length x, y and z. If triangle ABC  [#permalink]

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22 Feb 2020, 11:12
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 ?

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.
Re: Right triangle ABC has sides with length x, y and z. If triangle ABC   [#permalink] 22 Feb 2020, 11:12