geo data sufficiency

[guest612]

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

(1) The length of the line segment QR is equal to the length of line segment RS.

(2) The length of line segment ST is equal to the length of line segment TU.

Correct Answer is C. I thought it was E. I'm not sure how the lengths of line segments indicate the measurement of angles such that I can conclude the measurement for angle x. I understand there are two isosceles triangles as indicated by two statements. Are there three right isosceles triangles including the largest triangle as a whole? Even if so, how can I derive x?

Thanks!

[Sudhan]

As per the statements 1 and 2, we can form an another triangle by joining Q and C as mentioned in the diagram. so we will have 5 triangles, RPT, RQS, SCT, SQC and PQC. To find x, we need to know the measurements of triangle SQC. Angle x= QSC which is the hyptoneuse for triangle QPC which is right triangle.

Hence C.

Thanks

[guest621]

Thanks for your response Sudhan. I think I'm a little confused as to where C is. How can I join Q & C? Where is C? Is that where S is?

[Sudhan]

I am sorry. That should be U not C as I mentioned earlier.

Thanks

[RonPurewal]

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

(1) The length of the line segment QR is equal to the length of line segment RS.

(2) The length of line segment ST is equal to the length of line segment TU.

Correct Answer is C. I thought it was E. I'm not sure how the lengths of line segments indicate the measurement of angles such that I can conclude the measurement for angle x. I understand there are two isosceles triangles as indicated by two statements. Are there three right isosceles triangles including the largest triangle as a whole? Even if so, how can I derive x?

Thanks!

well, you've got two isosceles triangles if you take the two statements together. so, in other words, angles RQS and RSQ are the same, and angles TUS and TSU are the same.

the big triangle (the one containing everything in the problem) is a right triangle, so you know that angle R and angle T add to 90 degrees. therefore, let angle r be r degrees, and let angle T be (90 - r) degrees.

then
angle RSQ = (180 - r)/2
= 90 - (r/2) degrees
and
angle TSU = (180 - (90 - r))/2
= 45 + (r/2) degrees

so
x = 180 - RSQ - TSU
= 45 degrees.

--

this is also an excellent problem for picking numbers. you can pick any number of degrees you want for angle R (as long as it's acute, of course), and then let angle T be 90 minus that number of degrees. then work your way through the problem, knowing that you can use the isosceles triangles to figure out everything else in the problem.

if you do this with two or three sets of numbers, you'll notice that you get 45 degrees every time. coincidence? not likely.

[guest612]

thank you for that explanation. that was great.

[rfernandez]

[Guest]

This problem is insane. Things like this make me say to myself, "a person that would get a 790 would look at these two combined and come up with an answer in 30 seconds. I'm going with C."

[calgmatter]

Goal is to find x.
If we find ang(RSQ) and ang (UST), we can add them and subtract the result from 180 to get x.

ang (RSQ) = y and ang (STU) = z [say]

ang (RQS) = y and ang (SUT) = y [because RQS and SUT are isoceles]

In RQS, ang (QRS) = 180-2y and in SUT, ang (STU)=180-2z

In RPT, 90 + 180-2y + 180-2z = 180,
=> y+z = 135, => x=45

[RonPurewal]

This problem is insane. Things like this make me say to myself, "a person that would get a 790 would look at these two combined and come up with an answer in 30 seconds. I'm going with C."

if you're going to use this sort of reasoning when you guess**, be sure that you're not falling for the dreaded "c trap". for information on the "c trap", check out my post dated 24th july 2:25am on this thread. as of this writing it's the last post on that thread, but of course that may change.

**if you're using reasoning like this for anything other than desperate last-ditch guessing, that's a mistake.

how long would someone with an 800 take?
heh heh

[parthian7]

another approach:

let's call <RQS: b and <TSU: a

1. x + a + b = 180
and we know the interior angles in SQPU add up to 360. Therefore:

x + 90 + 180 - a + 180 - b = 360 --> a + b = x + 90

substitute in 1 --> 2x + 90 = 180 --> x=45

[jlucero]

Not only does it work, it's my own preferred method for these types of problems. Remember that you can always add variables to Geometry diagrams as long as you work to eliminate those variables with the extra information they give you. Well done!

[Peiyilin09]

I was able to figure out this problem after 10 minutes of thinking about it. How is it possible to figure this out in less than 3 min on the test? Should I just make educated guesses?

[tim]

sure. it should be obvious fairly quickly that neither statement alone is enough because that gives too much flexibility to the angle in question. start labeling everything you can right from the beginning. this is a problem about angles, so label the two acute angles in the right triangle q and 90-q. then you've got isosceles triangles so figure out what the other angles are in terms of q, and from that you should be able to get an expression for the angle you're looking for. when the q's drop out, you know you have sufficient information to get a value..