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GMAT PREP - Geometry

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Source: — Data Sufficiency |
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by manpsingh87 » Sun Feb 27, 2011 5:58 am
aleph777 wrote:Would love some help with this one... Can't figure out how to use the information given to plot additional values.
Thanks!
First Consider statement 1) LQPR =30; as triangle QPS and RPS are right angle triangles, therefore LQPS+LPQR=90;
also LRPS+LPRS=90

LQPS= LQPR+LRPS;

LQPR+LRPS+LPQR=90;
asLQPR=30;
LRPS+LPQR=60;----------1)
also, LRPS+LPRS = 90;-----------2)
Subtracting 1 and 2 we get LPRS-LPQR=30; hence statement 1 alone is sufficient to answer the question;

Statement 2; LPQR+LPRQ=150;
Also we know that in triangle PQR LPQR+LPRQ+LQPR=180;
therefore LQPR=30;

hence we can use the same method as use in statement 1 to find the desired answer. Therefore both statement alone is sufficient.
Last edited by manpsingh87 on Sun Feb 27, 2011 6:06 am, edited 1 time in total.
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by Anurag@Gurome » Sun Feb 27, 2011 6:04 am
Statement 1: In triangle PQR,
  • angle QPR + angle PQR = (180 - angle PRQ) = angle PRS
Hence, angle PRS - angle PQR = angle QPR = 30 degrees

Sufficient

Statement 2: In triangle PQR,
  • angle QPR + angle PQR + angle PRQ = 180 degrees
Hence, angle QPR = (180 - (angle PQR + angle PRQ)) degrees = (180 - 150) degrees = 30 degrees, i.e. we have same information as of statement 1.

Sufficient

The correct answer is D.
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by maihuna » Sun Feb 27, 2011 6:32 am
Let us label PQR as y, PRS as x
=> RPS = 90-x, QPS = 90-y
=> RPQ = SPQ-SPR = 90-y-90+x = x-y

Also, x-y answers the difference in angle of PRS and QPR as asked,

1. Given QPR=30 => x-y= 30 so answer

2. From given info, we can get QPR=180-150 = 30 so this also answer as above.

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by GMATGuruNY » Sun Feb 27, 2011 7:53 am

Image

In the figure shown, the measure of angle PRS is how many degrees greater than the measure of angle PQR?

(1) The measure of angle QPR is 30 degrees.
(2) The sum of the measures of angles PQR and PRQ is 150 degrees.
A very useful approach when a DS question asks for the value of an angle (or, in this case, the difference between 2 angles):

Plug in twice, following the rules of geometry.

Why twice? So that we can see what happens to the value of PRS-PQR.

If the value of PRS-PQR stays the same, the statement is sufficient.
If the value of PRS-PQR changes, the statement is insufficient.

As we plug in, we have to follow the rules of geometry. If angles are inside a triangle, their sum must be 180. If angles form a straight line, their sum must be 180.

Statement 1: QPR = 30 degrees

Image

The image above shows two combinations of angle measurements in which QPR=30. In each case, PRS-PQR=30. Since the value of PRS-PQR stays the same, sufficient.

Statement 2: PQR + PRQ = 150 degrees
The image used in Statement 1 shows two combinations of angle measurements in which PQR+PRQ =150. In each case, PRS-PQR=30. Since the value of PRS-PQR stays the same, sufficient.

The correct answer is D.
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by aleph777 » Sun Feb 27, 2011 7:59 am
Mitch, in your solution, when you say plug in twice, you mean just pick arbitrary numbers to fill out your additional angles based on the first angle given in the statements?
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by GMATGuruNY » Sun Feb 27, 2011 8:10 am
aleph777 wrote:Mitch, in your solution, when you say plug in twice, you mean just pick arbitrary numbers to fill out your additional angles based on the first angle given in the statements?
Not quite arbitrary. We need to plug in angle measurements that satisfy the rules of geometry and all the conditions given in the problem. Plugging in allows us to see how all the angles affect each other.

Here's another problem in which I would plug in for the angles:
An easy approach to Q10 in the OG12 (the star question) is to plug in for the angle measurements. We're being asked to find the sum of the angles measurements of the 5 points of the star. The key is to plug in values that follow the rules of geometry:

Image

Let's start with the most unusual shape, the pentagon inside the star. For any polygon with n sides, the sum of the interior angles = (n-2)*180. Thus, the sum of the angles inside the pentagon = (5-2)*180 = 540. There are 5 angles inside the pentagon. To make the math easy, let's plug in 540/5 = 108 for each interior angle.

Each of the adjacent angles must be 180-108 = 72 (see the picture), because the sum of angles that form a straight line must be 180.

There are 5 triangles around the outside of the star. The sum of the angles inside each of these triangles must be 180. This forces each point of the star to be 180-72-72 = 36.

Since the star has 5 points, the sum of the angle measurements of all 5 points is 5*36 = 180.
Much easier than the solution offered in the OG12.
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by kareem.firoz » Sun Feb 27, 2011 11:03 pm
In this question, st(1) and st(2) are essentially the same. Rather, each statement can be deduced from the other one.
In such questions, is it fine if only one statement is evaluated, and based on whether it is Sufficient or Not Sufficient, the answer could be marked as either Both statments alone are Sufficient, or Both statements together are Not Sufficient ?

Can this strategy be used for similar questions, to save time?


Anurag@Gurome wrote:Statement 1: In triangle PQR,
  • angle QPR + angle PQR = (180 - angle PRQ) = angle PRS
Hence, angle PRS - angle PQR = angle QPR = 30 degrees

Sufficient

Statement 2: In triangle PQR,
  • angle QPR + angle PQR + angle PRQ = 180 degrees
Hence, angle QPR = (180 - (angle PQR + angle PRQ)) degrees = (180 - 150) degrees = 30 degrees, i.e. we have same information as of statement 1.

Sufficient

The correct answer is D.
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by abhi0697 » Fri Jun 17, 2011 4:50 am
Answer: D

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by [email protected] » Mon Feb 13, 2012 12:15 am
The correct answer is D yes agreed... Thank you all of you... Though this was a geometry question but it involved simple algebra...
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by ArunangsuSahu » Fri Feb 17, 2012 12:52 pm
(D)
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by pappueshwar » Wed Feb 29, 2012 7:37 am
hi mitch,

have a basic doubt here:

A) if QPR is 30 how did u arrive at other angles that are in red colour in the image ?

i am confused as to how angles like 10, 25, 50, 100, 35 and 115 are arrived at ...



GMATGuruNY wrote:

Image

In the figure shown, the measure of angle PRS is how many degrees greater than the measure of angle PQR?

(1) The measure of angle QPR is 30 degrees.
(2) The sum of the measures of angles PQR and PRQ is 150 degrees.
A very useful approach when a DS question asks for the value of an angle (or, in this case, the difference between 2 angles):

Plug in twice, following the rules of geometry.

Why twice? So that we can see what happens to the value of PRS-PQR.

If the value of PRS-PQR stays the same, the statement is sufficient.
If the value of PRS-PQR changes, the statement is insufficient.

As we plug in, we have to follow the rules of geometry. If angles are inside a triangle, their sum must be 180. If angles form a straight line, their sum must be 180.

Statement 1: QPR = 30 degrees

Image

The image above shows two combinations of angle measurements in which QPR=30. In each case, PRS-PQR=30. Since the value of PRS-PQR stays the same, sufficient.

Statement 2: PQR + PRQ = 150 degrees
The image used in Statement 1 shows two combinations of angle measurements in which PQR+PRQ =150. In each case, PRS-PQR=30. Since the value of PRS-PQR stays the same, sufficient.

The correct answer is D.
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