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alexdallas
- Master | Next Rank: 500 Posts
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Hey there,
If you have problems attaching pictures, then feel free to PM me with the document and I'll see to it that it's attached.
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In the figure shown, what is the value of x?
Source: Beat The GMAT — Data Sufficiency |
This is a quite difficult geometry problem.
I will use < sign to represent angles.
Statement 1 or 2 alone is not sufficient because we can move the other point around to form different angles of x.
When the two statements are combined, it gets tricky. I have two approaches. One is more robust than the other but will take more time.
Approach 1:
Assume <R = 60, then you can calculate x. You will find x = 45.
Assume <R = 10, you still find x = 45.
Assume <R = 45, you get x = 45.
Here I am just trying different values of <R to see if the value of x changes or not. With three totally different cases, it is safe to assume x = 45.
Approach 2:
Geometrically show that x = 45 no matter what <R is.
We know:
<RSQ + x + <TSU = 180
x = 180 - (<RSQ + <TSU)
We need to figure out what is <RSU + <TSU by using all the given conditions in the problem.
<R + <RSQ + <RQS = <R + 2<RSQ = 180
<T + <TSU + <TUS = <T + 2<TSU = 180
Adding the two equations yields:
<R + <T + 2(<RSQ + <TSU) = 360
We are almost there. We know RPT is a right triangle. Then
<R + <T = 90
Therefore
90 + 2(<RSQ + <TSU) = 360
<RSQ + <TSU = 135
Then
x = 180 - (<RSQ + <TSU) = 45
C is your answer.
I will use < sign to represent angles.
Statement 1 or 2 alone is not sufficient because we can move the other point around to form different angles of x.
When the two statements are combined, it gets tricky. I have two approaches. One is more robust than the other but will take more time.
Approach 1:
Assume <R = 60, then you can calculate x. You will find x = 45.
Assume <R = 10, you still find x = 45.
Assume <R = 45, you get x = 45.
Here I am just trying different values of <R to see if the value of x changes or not. With three totally different cases, it is safe to assume x = 45.
Approach 2:
Geometrically show that x = 45 no matter what <R is.
We know:
<RSQ + x + <TSU = 180
x = 180 - (<RSQ + <TSU)
We need to figure out what is <RSU + <TSU by using all the given conditions in the problem.
<R + <RSQ + <RQS = <R + 2<RSQ = 180
<T + <TSU + <TUS = <T + 2<TSU = 180
Adding the two equations yields:
<R + <T + 2(<RSQ + <TSU) = 360
We are almost there. We know RPT is a right triangle. Then
<R + <T = 90
Therefore
90 + 2(<RSQ + <TSU) = 360
<RSQ + <TSU = 135
Then
x = 180 - (<RSQ + <TSU) = 45
C is your answer.
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From statement 1) i.e. QR=RS
<RSQ = <RQS
From statement 2) i.e. ST=TU
<TSU = <SUT.
From big triangle PRT,
<R + <T = 90.
From statement 1) <R = 180-2 <RSQ.
From statement 2) <T = 180- 2 < TSU.
Hence (180-2 <RSQ ) + (180- 2 < TSU) = 90.
360-2( RSQ+ TSU ) =90
Now TRS forms a straight line. hence RSQ + X + TSU =180
Therefore,
RSQ + TSU = 180-X.
Therefore,
360-2 ( 180-x) =90
therefore,
x=45.
Hence from the equation it is clear that both the statements are necessary and enough to solve the problem.
<RSQ = <RQS
From statement 2) i.e. ST=TU
<TSU = <SUT.
From big triangle PRT,
<R + <T = 90.
From statement 1) <R = 180-2 <RSQ.
From statement 2) <T = 180- 2 < TSU.
Hence (180-2 <RSQ ) + (180- 2 < TSU) = 90.
360-2( RSQ+ TSU ) =90
Now TRS forms a straight line. hence RSQ + X + TSU =180
Therefore,
RSQ + TSU = 180-X.
Therefore,
360-2 ( 180-x) =90
therefore,
x=45.
Hence from the equation it is clear that both the statements are necessary and enough to solve the problem.
Hi GMATQuantCoach or any other senior people... can you provide your opinion on the way I solved it.. is it fine and refined??
It's good!
In these complicated geometry problems, just be sure to use all the given information. Most likely you will need all of them to solve the question.
Here, the given information is: RSQ and STU are isosceles triangles; RPT is a right triangle.
You were able to use all the information.
In these complicated geometry problems, just be sure to use all the given information. Most likely you will need all of them to solve the question.
Here, the given information is: RSQ and STU are isosceles triangles; RPT is a right triangle.
You were able to use all the information.
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Upcoming FREE Online Class: Effective Calculations - 11/15/09 Sunday
Instructor | GMATPrepMath.com | GMAT Prep Courses Starting at $60 with FREE GMAT Math Formula Sheet
Choose from Topics:
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Upcoming FREE Online Class: Effective Calculations - 11/15/09 Sunday
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Choose from Topics:
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Hi i am sowree i still dint get it!
Could someone help me explain this sum ?
I marked E coz the question is asking us to find the value of x = a solid #
How can we get a value from either or both of these statements ??
OA C
Could someone help me explain this sum ?
I marked E coz the question is asking us to find the value of x = a solid #
How can we get a value from either or both of these statements ??
OA C
Combining the statementsbhumika.k.shah wrote:Hi i am sowree i still dint get it!
Could someone help me explain this sum ?
I marked E coz the question is asking us to find the value of x = a solid #
How can we get a value from either or both of these statements ??
OA C
QRS+RTP =90 ---- (1)
UST+x+QSR =180 ---- (2)
QRS+2QSR =180 - (3) (QSR = RQS and sum of the angles of the triangle QSR =180)
RTP+ 2UST =180 ---(4) (UST = SUT and sum of the angles of the triangle UST=180)
QRS+RTP +2(QSR+UST) =360 ((3)+(4))
QSR+UST = 135 ---(5)
X =45 ((5) in 2)
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Mom4MBA
- Senior | Next Rank: 100 Posts
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after checking that each statement alone will not do any good we will check both the statements together. so use the basic properties or triangles and quadrilaterals and start working.
take the big triangle:
sum of the three angles = 180
180-2a + 180-2b + 90 = 180
a+b = 135
now take the quadrilateral:
sum of the 4 angles = 360
180-a + 180-b + x + 90 = 360
x= a+b - 90
x= 135 - 90
x= 45[/img][/url]
take the big triangle:
sum of the three angles = 180
180-2a + 180-2b + 90 = 180
a+b = 135
now take the quadrilateral:
sum of the 4 angles = 360
180-a + 180-b + x + 90 = 360
x= a+b - 90
x= 135 - 90
x= 45[/img][/url]
Thanks!
finally got it this way
finally got it this way
Mom4MBA wrote:after checking that each statement alone will not do any good we will check both the statements together. so use the basic properties or triangles and quadrilaterals and start working.
take the big triangle:
sum of the three angles = 180
180-2a + 180-2b + 90 = 180
a+b = 135
now take the quadrilateral:
sum of the 4 angles = 360
180-a + 180-b + x + 90 = 360
x= a+b - 90
x= 135 - 90
x= 45[/img][/url]
