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Source: — Data Sufficiency |

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by Anurag@Gurome » Wed Jan 12, 2011 1:33 am
tatser@09 wrote:In the figure , what is the value of x?

1. Length of QR = length of RS
2. Length of ST = Length of TU
Note that the measure of angle x depends upon the position of points Q, S and U only. Unless we don't know the fixed positions of these three points, we cannot uniquely determine the measure of angle x.

Statement 1: QR = RS
Thus position of Q and S is fixed. But U can be any point on PT and accordingly value of x will be different.

Not sufficient

Statement 2: ST = TU
Thus position of S and U is fixed. But Q can be any point on PR and accordingly value of x will be different.

Not sufficient

1 & 2 Together: Now the three points are fixed. Let's see whether we can find x. Refer to the image below.
Image

On point S, the sum of the three angles must be equal to 180°.
Thus, (x + y + z) = 180° ..................................... (i)

angle PQS = (180° - angle RQS) = (180° - z)
angle PUS = (180° - angle TUS) = (180° - y)

Now in quadrilateral PQSU,
  • Sum of all the internal angles = 360°
    => [x + 90° + (180° - y) + (180° - z)] = 360°
    => (x - y - z + 90°) = 0 .................................. (ii)
Now add (i) and (ii) => (2x + 90°) = 180° => x = 45°

Sufficient

The correct answer is C.
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by tatser@09 » Wed Jan 12, 2011 2:32 am
Anurag@Gurome wrote:
tatser@09 wrote:In the figure , what is the value of x?

1. Length of QR = length of RS
2. Length of ST = Length of TU
Note that the measure of angle x depends upon the position of points Q, S and U only. Unless we don't know the fixed positions of these three points, we cannot uniquely determine the measure of angle x.

Statement 1: QR = RS
Thus position of Q and S is fixed. But U can be any point on PT and accordingly value of x will be different.

Not sufficient

Statement 2: ST = TU
Thus position of S and U is fixed. But Q can be any point on PR and accordingly value of x will be different.

Not sufficient

1 & 2 Together: Now the three points are fixed. Let's see whether we can find x. Refer to the image below.
Image

On point S, the sum of the three angles must be equal to 180°.
Thus, (x + y + z) = 180° ..................................... (i)

angle PQS = (180° - angle RQS) = (180° - z)
angle PUS = (180° - angle TUS) = (180° - y)

Now in quadrilateral PQSU,
  • Sum of all the internal angles = 360°
    => [x + 90° + (180° - y) + (180° - z)] = 360°
    => (x - y - z + 90°) = 0 .................................. (ii)
Now add (i) and (ii) => (2x + 90°) = 180° => x = 45°

Sufficient

The correct answer is C.
Understanding the problem itself took me more time , I am worried as to how long does these problems take during the exam :(
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by GMATGuruNY » Wed Jan 12, 2011 4:07 am
haidgmat wrote:Image

In the figure shown, what is the value of X?

1. QR=RS

2. ST=TU
Neither statement alone is sufficient to determine the value of x. When we combine the two statements, here's an efficient way to determine whether we have sufficient information:

1. Plug in values for all the angle measurements, following the conditions in the problem and the rules of geometry.
2. Determine the value of x.
3. Plug in different values for all the angle measurements, still following the conditions in the problem and the rules of geometry.
3. Determine the value of x.

If the value of x stays the same, we have sufficient information.
If the value of x changes, we have insufficient information.

The image below shows two sets of angle measurements that satisfy both the rules of geometry and the information given in the two statements:

∠PRT + ∠PTR = 90 because triangle PRT is a right triangle.
Since QR=RS, ∠RQS = ∠RSQ.
Since ST=TU, ∠UST = ∠SUT.
Since the sum of angles that form a straight line is 180, x = 180 - ∠RQS - ∠UST.

Image

In each case, x=45. Thus, when the two statements are combined, we know that x=45 and that the two statements combined are sufficient.

The correct answer is C.
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