Find all School-related info fast with the new School-Specific MBA Forum

 It is currently 27 Jul 2016, 21:09

### GMAT Club Daily Prep

#### Thank you for using the timer - this advanced tool can estimate your performance and suggest more practice questions. We have subscribed you to Daily Prep Questions via email.

Customized
for You

we will pick new questions that match your level based on your Timer History

Track

every week, we’ll send you an estimated GMAT score based on your performance

Practice
Pays

we will pick new questions that match your level based on your Timer History

# Events & Promotions

###### Events & Promotions in June
Open Detailed Calendar

# In the diagram to the right, triangle PQR has a right angle

Author Message
TAGS:

### Hide Tags

Manager
Joined: 19 Aug 2007
Posts: 169
Followers: 1

Kudos [?]: 40 [1] , given: 0

In the diagram to the right, triangle PQR has a right angle [#permalink]

### Show Tags

31 Oct 2007, 09:07
1
KUDOS
5
This post was
BOOKMARKED
00:00

Difficulty:

85% (hard)

Question Stats:

64% (04:29) correct 36% (03:34) wrong based on 140 sessions

### HideShow timer Statistics

In the diagram to the right, triangle PQR has a right angle at Q and a perimeter of 60. Line segment QS is perpendicular to PR and has a length of 12. PQ > QR. What is the ratio of the area of triangle PQS to the area of triangle RQS?

A.. 3/2
B. 7/4
C. 15/8
D. 16/9
E. 2
Attachment:

triangle.jpg [ 6.39 KiB | Viewed 4445 times ]

OPEN DISCUSSION OF THIS QUESTION IS HERE: in-the-diagram-triangle-pqr-has-a-right-angle-at-q-and-a-127093.html
[Reveal] Spoiler: OA

Last edited by Bunuel on 16 Apr 2012, 00:46, edited 1 time in total.
Edited the question and added the OA. Topic is locked.
Director
Joined: 11 Jun 2007
Posts: 931
Followers: 1

Kudos [?]: 139 [2] , given: 0

### Show Tags

31 Oct 2007, 09:47
2
KUDOS
1
This post was
BOOKMARKED
jimjohn wrote:
In the diagram to the right, triangle PQR has a right angle at Q and a perimeter of 60. Line segment QS is perpendicular to PR and has a length of 12. PQ > QR. What is the ratio of the area of triangle PQS to the area of triangle RQS?

3/2

7/4

15/8

16/9

2

i get 16/9 but after spending too much time trying to figure it out.
i'll try my best to explain as best as i can (see my diagram below)

stem only tells you 1) these three triangles are all right triangles (so you can easily apply pythagorean theorem), 2) the perimeter of the largest triangle, 3) the length of the bisector, and 4) PQ > QR. From this you have to think about the possible lengths to find the sides.

I thought about the common right triangle sides: 3-4-5, 6-8-10, etc and saw that 3-4-5 = 12 and 60 is a multiple of 12 (5x). So the possible sides are [3-4-5]*5 = 15-20-25. But first have to test it out. We know that PQ > QR and the largest side is the hypotenuse. So PQ = 20, QR = 15, and PR = 25. From there I used side QS = 12 to figure out the splits between PS and RS. (Lucky thing it worked out!!)

PS^2 + QS^2 = PQ^2
PS^2 = PQ^2 - QS^2
400 - 144 = 256
sqrt256 = 16

QS^2 + RS^2 = QR^2
RS^2 = QR^2 - QS^2
225 - 144 = 81
sqrt 81 = 9

so the ratio of the area of triangle PQS to the area of triangle RQS is
PQS = 1/2bh
1/2 * 12 * 16 = 96

RQS = 1/2bh
1/2 * 12 * 9 = 54

96:54 = 16:9
Attachments

geometry.JPG [ 9.88 KiB | Viewed 4400 times ]

Last edited by beckee529 on 31 Oct 2007, 09:54, edited 2 times in total.
Current Student
Joined: 18 Jun 2007
Posts: 408
Location: Atlanta, GA
Schools: Emory class of 2010
Followers: 11

Kudos [?]: 40 [0], given: 0

### Show Tags

31 Oct 2007, 09:53
Yeah, and actually, your calculations were still correct. I made an error in my calculations, the answer is not 2. It is 16/9 as you stated.
Director
Joined: 11 Jun 2007
Posts: 931
Followers: 1

Kudos [?]: 139 [0], given: 0

### Show Tags

31 Oct 2007, 09:55
emoryhopeful wrote:
Yeah, and actually, your calculations were still correct. I made an error in my calculations, the answer is not 2. It is 16/9 as you stated.

yeah i had it right on paper but wrote it incorrectly on the computer screen but have since edited to correct it
Senior Manager
Joined: 28 Jun 2009
Posts: 454
Location: United States (MA)
Followers: 18

Kudos [?]: 147 [0], given: 46

Re: In the diagram to the right, triangle PQR has a right angle [#permalink]

### Show Tags

15 Apr 2012, 18:27

we've given that the hypotenuse of triangle PQR is 25.
Applying Pythagorean triplet 15-20-25,

PQ and QR are respectively 20 and 15.

So area = 1/2 * base QR * height PQ
= 1/2 * 15 * 20
= 150
Manager
Status: I will not stop until i realise my goal which is my dream too
Joined: 25 Feb 2010
Posts: 235
Schools: Johnson '15
Followers: 2

Kudos [?]: 48 [0], given: 16

Re: In the diagram to the right, triangle PQR has a right angle [#permalink]

### Show Tags

15 Apr 2012, 21:52
jimjohn wrote:
In the diagram to the right, triangle PQR has a right angle at Q and a perimeter of 60. Line segment QS is perpendicular to PR and has a length of 12. PQ > QR. What is the ratio of the area of triangle PQS to the area of triangle RQS?

3/2

7/4

15/8

16/9

2

can somebody give OA for this?
_________________

Regards,
Harsha

Note: Give me kudos if my approach is right , else help me understand where i am missing.. I want to bell the GMAT Cat

Satyameva Jayate - Truth alone triumphs

Math Expert
Joined: 02 Sep 2009
Posts: 34092
Followers: 6095

Kudos [?]: 76669 [6] , given: 9978

Re: In the diagram to the right, triangle PQR has a right angle [#permalink]

### Show Tags

16 Apr 2012, 00:45
6
KUDOS
Expert's post
2
This post was
BOOKMARKED
harshavmrg wrote:
can somebody give OA for this?

In the diagram, triangle PQR has a right angle at Q and a perimeter of 60. Line segment QS is perpendicular to PR and has a length of 12. PQ > QR. What is the ratio of the area of triangle PQS to the area of triangle RQS?
A. 3/2
B. 7/4
C. 15/8
D. 16/9
E. 2
Attachment:

Triangle PQR.GIF [ 2.52 KiB | Viewed 4169 times ]

Let $$PQ=x$$, $$QR=y$$ and $$PR=z$$.

Given: $$x+y+z=60$$ (i);
Equate the areas: $$\frac{1}{2}*xy=\frac{1}{2}*QS*z$$ (area of PQR can be calculated by 1/2*leg*leg and 1/2* perpendicular to hypotenuse*hypotenuse) --> $$xy=12z$$ (ii);
Aslo $$x^2+y^2=z^2$$ (iii);

So, we have:
(i) $$x+y+z=60$$;
(ii) $$xy=12z$$;
(iii) $$x^2+y^2=z^2$$.

From (iii) $$(x+y)^2-2xy=z^2$$ --> as from (i) $$x+y=60-z$$ and from (ii) $$xy=12z$$ then ($$60-z)^2-2*12z=z^2$$ --> $$3600-120z+z^2-24z=z^2$$ --> $$3600=144z$$ --> $$z=25$$;

From (i) $$x+y=35$$ and from (ii) $$xy=300$$ --> solving for $$x$$ and $$y$$ --> $$x=20$$ and $$y=15$$ (as given that $$x>y$$).

Next, perpendicular to the hypotenuse will always divide the triangle into two triangles with the same properties as the original triangle. So, PQR and SQR are similar. In two similar triangles, the ratio of their areas is the square of the ratio of their sides: $$\frac{AREA}{area}=\frac{S^2}{s^2}$$.

So, $$\frac{x^2}{y^2}=\frac{AREA}{area}$$ --> $$\frac{AREA}{area}=\frac{400}{225}=\frac{16}{9}$$

OPEN DISCUSSION OF THIS QUESTION IS HERE: in-the-diagram-triangle-pqr-has-a-right-angle-at-q-and-a-127093.html
_________________
Re: In the diagram to the right, triangle PQR has a right angle   [#permalink] 16 Apr 2012, 00:45
Similar topics Replies Last post
Similar
Topics:
7 In the diagram to the right, equilateral triangle ADE is 9 28 Jul 2014, 05:21
18 In the diagram to the right, triangle PQR has a right angle 12 29 Mar 2014, 00:17
10 In the diagram above, <PQR is a right angle, and QS is 13 05 May 2012, 19:37
63 In the diagram, triangle PQR has a right angle at Q and a 27 05 Feb 2012, 16:54
18 Triangle PQR is right angled at Q. QT is perpendicular to PR 13 12 Dec 2010, 06:28
Display posts from previous: Sort by