In the diagram below, what is the length of RS?

Question

In the diagram below, what is the length of RS? https://gmatclub.com/forum/download/file.php?id=36331 (1) PQ = 28 and TU = 21. (2) PQ, RS and TU are parallel lines. 2017-07-18_1022.png

jedit · Answer

It appears to be C.

Statement 1: Insufficient.

We do not know the actual height from the base.

Statement 2: Insufficient.

We do not know lengths.

Combining, if the lines are parallel, their intersection should be the half of the average of the two lengths.

mohshu · Answer

to me it luks like E...
the two big triangles are similar and we will get the ratio not the absolute values,....
any help on this is appreciated..
thanks

Alexey1989x · Answer

I presume that it's E, because from (1) + (2) we cannot deduce whether RS is the mid-section of triangles or not. 
PQ/TU=4/3 hence we have 1 and 2 for RS, and it is still possible for RS to be 7 or 14. 
It also can be visualized if we make it wider ot thinner the overlapping of 2 triangles PQU & TQU.

sonikavadhera · Answer

please explain how you came to the conclusion.

RK007 · Answer

Is this a theorem? Do u have any explaination?

RK007 · Answer

Were you able to find an answer to this?

NCRanjan · Answer

Hmm well its a nice question,

I chose E thinking on the lines that the only way C can be the answer would be if the bigger triangles are also similar ie if PQU is similar to TQU

Stmt 1 - only gives the value of PQ and TU and tells that they are not congruent. Fair enough
Stmt 2 says - RUS is similar to PUQ and that RQS is similar to TQU . no dimensions no answer

so if statement two also means PUQ and TUQ similar then we have our answer
RS/21= RS/28

RS=1/7
i want to believe it but since i am not sure i wont advocate it and wait for some expert to comment on it

JeffYin · Answer

I'm not going to worry about whether this is a theorem or not, since approaching this problem from basic principles will be more relevant for the GMAT anyway.

I would start by re-drawing the diagram, and labeling the lengths of segments with variables as I have done below:

https://yintutoring.com/wp-content/uploads/2018/12/Similar-triangle-diagram.png

Note that we are solving for c in the above diagram. Now, based on statement 2, if we see that triangle RSU is similar to triangle PQR, and triangle RSQ is similar to triangle TUQ, we can write the ratios of sides as the following equations:

\frac{c}{e} = \frac{a}{d+e}  and  \frac{c}{d} = \frac{b}{d+e}

If we cross-multiply these equation we get:

c(d+e) = ae  and  c(d+e) = bd

This tells us that:

ae=bd

Now, statement 1 tells us that a = 28 and b = 21, which turns the equation above into:

28e = 21d

Thus, we have the ratio of e to d, which means that we also can calculate the ratio of e to d+e. We can rearrange the equation  c(d+e) = ae  to be:

c = a*\frac{e}{d+e}

Finally, since we know a = 28 and we can calculate the ratio  \frac{e}{d+e} , this is sufficient to calculate c, and thus answer the question.

Please let me know if you have any questions!

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In the diagram below, what is the length of RS?

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In the diagram below, what is the length of RS?

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