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

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If PQ = 1, what is the length of RS ?  [#permalink]

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New post 02 Jul 2018, 23:59
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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. \(\frac{2}{\sqrt{12}}\)


Attachment:
triangle.jpg
triangle.jpg [ 19.78 KiB | Viewed 737 times ]

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If PQ = 1, what is the length of RS ?  [#permalink]

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New post Updated on: 03 Jul 2018, 01:24
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Bunuel wrote:
Image
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. \(\frac{2}{\sqrt{12}}\)


Attachment:
triangle.jpg


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.
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Originally posted by DavidTutorexamPAL on 03 Jul 2018, 00:22.
Last edited by DavidTutorexamPAL on 03 Jul 2018, 01:24, edited 1 time in total.
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Re: If PQ = 1, what is the length of RS ?  [#permalink]

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New post 03 Jul 2018, 01:21
DavidTutorexamPAL wrote:
Bunuel wrote:
Image
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. \(\frac{2}{\sqrt{12}}\)


Attachment:
triangle.jpg


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))

Bunuel are the answers correct?


I think so. Try to rationalize RS = 1/(4*sqrt(3)).
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Collection of Questions:
PS: 1. Tough and Tricky questions; 2. Hard questions; 3. Hard questions part 2; 4. Standard deviation; 5. Tough Problem Solving Questions With Solutions; 6. Probability and Combinations Questions With Solutions; 7 Tough and tricky exponents and roots questions; 8 12 Easy Pieces (or not?); 9 Bakers' Dozen; 10 Algebra set. ,11 Mixed Questions, 12 Fresh Meat

DS: 1. DS tough questions; 2. DS tough questions part 2; 3. DS tough questions part 3; 4. DS Standard deviation; 5. Inequalities; 6. 700+ GMAT Data Sufficiency Questions With Explanations; 7 Tough and tricky exponents and roots questions; 8 The Discreet Charm of the DS; 9 Devil's Dozen!!!; 10 Number Properties set., 11 New DS set.


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If PQ = 1, what is the length of RS ?  [#permalink]

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New post 03 Jul 2018, 01:25
Bunuel wrote:
DavidTutorexamPAL wrote:
Bunuel are the answers correct?


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


Ah, didn't think to do that. Thanks :)
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Re: If PQ = 1, what is the length of RS ?  [#permalink]

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New post 03 Jul 2018, 01:58
Option B using 30-60-90
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Re: If PQ = 1, what is the length of RS ?  [#permalink]

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New post 03 Jul 2018, 12:09
can we use the formula:- QS * RS=TS^2 ?
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If PQ = 1, what is the length of RS ?  [#permalink]

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New post 03 Jul 2018, 19:17
Bunuel wrote:
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. \(\frac{2}{\sqrt{12}}\)

Attachment:
triangle2018.7.3.jpg
triangle2018.7.3.jpg [ 29.28 KiB | Viewed 471 times ]

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
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If PQ = 1, what is the length of RS ? &nbs [#permalink] 03 Jul 2018, 19:17
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