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

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Math Expert
Joined: 02 Sep 2009
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If PQ = 1, what is the length of RS ?  [#permalink]

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02 Jul 2018, 23:59
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45% (medium)

Question Stats:

71% (02:31) correct 29% (03:17) wrong based on 49 sessions

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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 [ 19.78 KiB | Viewed 737 times ]

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

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Updated on: 03 Jul 2018, 01:24
1
1
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:
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.

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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.
Math Expert
Joined: 02 Sep 2009
Posts: 49430
Re: If PQ = 1, what is the length of RS ?  [#permalink]

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03 Jul 2018, 01:21
DavidTutorexamPAL wrote:
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:
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))

I think so. Try to rationalize RS = 1/(4*sqrt(3)).
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If PQ = 1, what is the length of RS ?  [#permalink]

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03 Jul 2018, 01:25
Bunuel wrote:
DavidTutorexamPAL wrote:

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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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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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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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 [ 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

$$x = (\frac{1}{4\sqrt{3}} * \frac{\sqrt{3}}{\sqrt{3}})$$

$$x= RS = \frac{\sqrt{3}}{12}$$

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