If PQ = 1, what is the length of RS ?

Question

https://gmatclub.com/forum/download/file.php?id=41968 If PQ = 1, what is the length of RS ? A. \frac{1}{12} B. \frac{\sqrt{3}}{12} C. \frac{1}{6} D. \frac{2}{3 \sqrt{3}} E. \frac{2}{\sqrt{12}} triangle.jpg

DavidTutorexamPAL · Answer

Questions involving right-angle triangles can often be solved by applying Pythagorean thm or special-angle rules.
We'll look for these rules, a Precise approach.

Since  \angle QPT = 30, then  \angle QST=60 and we can also fill in  \angle RTS=30 and  \angle RQT =30. That is, all our triangles are 30-60-90 triangles.
This means that the ratio between the short leg and the hypotenuse is 1:2 and the long leg to the short leg is sqrt(3):1.
So: 
PQ = 1 --> QT = 1/2
QT = 1/2 --> RT = 1/4
RT = 1/4 --> RS = 1/(4*sqrt(3)) = sqrt(3)/sqrt(3) * 1/(4*sqrt(3)) = sqrt(3)/12.

(B) is our answer.

Bunuel · Answer

I think so. Try to rationalize RS = 1/(4*sqrt(3)).

DavidTutorexamPAL · Answer

Ah, didn't think to do that. Thanks

pandeyashwin · Answer

Option B using 30-60-90

PKGMAT · Answer

can we use the formula:- QS * RS=TS^2 ?

generis · Answer

triangle2018.7.3.jpg Calculations here are quite simple because all the triangles are 30-60-90 Sides opposite those angles are in ratio x : x\sqrt{3} : 2x We are given that the angle at vertex P = 30°. The angle at P starts a chain in which we find nothing except 30-60-90 angle possibilities ∠PTQ = 90°. The third angle must = 60° (∠PQT) In turn, that 60° is part of a 90° angle, so adjacent ∠SQT = 30° In turn, ∆QRT 's second angle = 90° . . . etc. Halve two sides (PQ and QT), divide the third (RT) by \sqrt{3} , and we have RS. (1) For ∆ PQS, side PQ = 1 PQ, opposite the 90° angle, corresponds with 2x in the ratio of sides 2x = 1 x = \frac{1}{2} x corresponds with QT, the side opposite the 30° angle. QT = \frac{1}{2} (2) For ∆ QST, side QT = \frac{1}{2} QT, opposite the 90° angle, corresponds with 2x RT, opposite the 30° angle, corresponds with x So RT = \frac{1}{2}QT QT = \frac{1}{2} RT =( \frac{1}{2}* \frac{1}{2}) = \frac{1}{4} (3) For ∆, side RT = \frac{1}{4} RT, opposite the 60° angle, corresponds with x\sqrt{3} RT = \frac{1}{4} = x\sqrt{3} RS = x = \frac{1}{4\sqrt{3}}= RS, opposite the 30° angle x = (\frac{1}{4\sqrt{3}} * \frac{\sqrt{3}}{\sqrt{3}}) x= RS = \frac{\sqrt{3}}{12} Answer B

dcummins · Answer

All 3 triangles are 30-60-90.

And they are all similar.

We can find the length of one more side in the larger triangle to determine the similar triangle ratio and one side of the triangle QTR. We can then use the properties of similar triangles to determine the side RS

RastogiSarthak99 · Answer

Obviously everyone's explanations are straightforward. I'll offer my perspective on how to go about solving:

If you've cracked that all angles in 3 triangles are 30-60-90, then it's just a matter of finding values for all sides one by one. Don't get distracted, and keep going till you find RS which is the side opposite to a 30 degree angle.

B

If PQ = 1, what is the length of RS ?

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If PQ = 1, what is the length of RS ?

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