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# What is the perimeter of ∆PQS ?

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What is the perimeter of ∆PQS ?  [#permalink]

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Updated on: 02 Oct 2015, 10:40
1
3
00:00

Difficulty:

95% (hard)

Question Stats:

34% (01:51) correct 66% (03:07) wrong based on 91 sessions

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What is the perimeter of ∆PQS ?

(1) x = 45
(2) w = 15

Spoiler: :: OE
Because angle w is outside of ∆PQS, it is tempting to think that angle w is not required to construct ∆PQS,
but it is. The second paragraph below shows that w can determine x. Hence, if Statement (1) alone can
answer the question, then Statement (2) can as well.
The minimum details needed to construct a triangle are one side and two angles, or two sides and one
angle, or all the three sides. In ∆PQS, we have ∠PSQ = 30° (from the figure) and the side PS = 1. So, we
are short of knowing one angle or one side to construct ∆PQS. Statement (1) would help with an angle in
the triangle. So, Statement (1) is sufficient.
Construct line PS = 1. Draw an angle making 30 degrees with PS at S and name the line l. Extend the line
PS 2 units further to the right of the point S to locate the point R. Now, draw another line measuring w (=
15) degrees with PR from the point R, and name the line m. So, the point of intersection of l and m is the
point Q. Now, measure ∠PQS to find x.

Attachment:

Image.png [ 19.1 KiB | Viewed 1605 times ]

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Originally posted by arhumsid on 28 Sep 2015, 19:56.
Last edited by arhumsid on 02 Oct 2015, 10:40, edited 4 times in total.
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Re: What is the perimeter of ∆PQS ?  [#permalink]

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28 Sep 2015, 20:24
is it E? QP cannot be found?
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Re: What is the perimeter of ∆PQS ?  [#permalink]

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29 Sep 2015, 23:59
This question is from "Nova's GMAT Data Sufficiency Prep Course" page #54. Fed up with my pathetic performance on DS, i came across this DS-book and its completely different approach to DS problems (i never thought of DS problems the way they take them. No comments of effectiveness though).

As far as this problem is concerned, we dont have to 'find' the perimeter, what we need to know is whether it can be found.

Like every other Quant question, for this question too i am waiting for Bunuel to reply.

Below is the OE by Nova though

Spoiler: :: OE
Because angle w is outside of ∆PQS, it is tempting to think that angle w is not required to construct ∆PQS,
but it is. The second paragraph below shows that w can determine x. Hence, if Statement (1) alone can
answer the question, then Statement (2) can as well.
The minimum details needed to construct a triangle are one side and two angles, or two sides and one
angle, or all the three sides. In ∆PQS, we have ∠PSQ = 30° (from the figure) and the side PS = 1. So, we
are short of knowing one angle or one side to construct ∆PQS. Statement (1) would help with an angle in
the triangle. So, Statement (1) is sufficient.
Construct line PS = 1. Draw an angle making 30 degrees with PS at S and name the line l. Extend the line
PS 2 units further to the right of the point S to locate the point R. Now, draw another line measuring w (=
15) degrees with PR from the point R, and name the line m. So, the point of intersection of l and m is the
point Q. Now, measure ∠PQS to find x.

_________________
One Kudos for an everlasting piece of knowledge is not a bad deal at all...

------------------------------------------------------------------------------------------------------------------------
Twenty years from now you will be more disappointed by the things you didn't do than by the ones you did do. So throw off the bowlines. Sail away from the safe harbor. Catch the trade winds in your sails. Explore. Dream. Discover.
-Mark Twain
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Re: What is the perimeter of ∆PQS ?  [#permalink]

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12 Oct 2015, 14:33
arhumsid wrote:

What is the perimeter of ∆PQS ?

(1) x = 45
(2) w = 15

Spoiler: :: OE
Because angle w is outside of ∆PQS, it is tempting to think that angle w is not required to construct ∆PQS,
but it is. The second paragraph below shows that w can determine x. Hence, if Statement (1) alone can
answer the question, then Statement (2) can as well.
The minimum details needed to construct a triangle are one side and two angles, or two sides and one
angle, or all the three sides. In ∆PQS, we have ∠PSQ = 30° (from the figure) and the side PS = 1. So, we
are short of knowing one angle or one side to construct ∆PQS. Statement (1) would help with an angle in
the triangle. So, Statement (1) is sufficient.
Construct line PS = 1. Draw an angle making 30 degrees with PS at S and name the line l. Extend the line
PS 2 units further to the right of the point S to locate the point R. Now, draw another line measuring w (=
15) degrees with PR from the point R, and name the line m. So, the point of intersection of l and m is the
point Q. Now, measure ∠PQS to find x.

I will try,

The Question is whether we can draw a fixed Triangle ∆PQS or not.

As we know to get a fixed Triangle, we need atleast two sides and one included angle or two angles and one included side.

Statement 1): We are given ∠x=45 then ∠p=180-45-30=105.
Now we have Two angles and one included side so we have fixed triangle.
we will have fixed perimeter for ∆PQS.

Hence , Sufficient.

Statement 2): We are given ∠w=15 and also we know ∠y=150, so ∠(q-x)=15=∠w.
now ∆SQR is Isosceles trainlge, if side opposite to ∠(q-x) ,SR, is 2 mm then side opposite to ∠w, QS, is also 2 mm.
Now in ∆PQS, we have two sides and one included angle , so we have one fixed trainlge ∆PQS.
we will have fixed perimeter for ∆PQS.

Hence, Suffiecient.

And) D

please correct me if i am wrong.
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Re: What is the perimeter of ∆PQS ?  [#permalink]

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12 Oct 2015, 15:20
arhumsid wrote:

What is the perimeter of ∆PQS ?

(1) x = 45
(2) w = 15

Spoiler: :: OE
Because angle w is outside of ∆PQS, it is tempting to think that angle w is not required to construct ∆PQS,
but it is. The second paragraph below shows that w can determine x. Hence, if Statement (1) alone can
answer the question, then Statement (2) can as well.
The minimum details needed to construct a triangle are one side and two angles, or two sides and one
angle, or all the three sides. In ∆PQS, we have ∠PSQ = 30° (from the figure) and the side PS = 1. So, we
are short of knowing one angle or one side to construct ∆PQS. Statement (1) would help with an angle in
the triangle. So, Statement (1) is sufficient.
Construct line PS = 1. Draw an angle making 30 degrees with PS at S and name the line l. Extend the line
PS 2 units further to the right of the point S to locate the point R. Now, draw another line measuring w (=
15) degrees with PR from the point R, and name the line m. So, the point of intersection of l and m is the
point Q. Now, measure ∠PQS to find x.

I think Nova is a good book to practice but the questions sometimes can be a bit too calculation intensive or ask to apply very specific formulae. This is one such question.

Statement 1 is sufficient as mentioned in posts above.

For statement 2, once you figure out that QSR is an isosceles triangle giving you QS=2. You are already given PS=1 and $$\angle {QSP} = 30$$.

Remember the "rule" that you can create a fixed triangle with 2 sides and the included angle.

But for the proof of it, you will have to use 3 trigonometric formulae (not recommended for GMAT)

1. $$sin^2(x) + cos^2(x) = 1$$
2.$$sin (x+y)= sin(x)*cos(y)+sin(y)*cos(x)$$
3. $$a/sin(x)=b/sin(y)=c/sin(z)$$

Rest assured you will be able to find unique value for PQ leading to a unique perimeter for triangle PQS.

Hope this helps.
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Re: What is the perimeter of ∆PQS ?  [#permalink]

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10 Nov 2015, 17:18
trigonometry helps a a lot:

(1) x=45

the area of a triangle equals 1/2*a*b*sinA

since you know all the three angles and one line, all the length of three lines can be obtained:

1: 1/2*PQ*1*sin105=1/2*PQ*QS*sin45

so you have QS

then 1/2*QS*1*sin30=1/2*PQ*QS*sin45

so you have PQ

sufficient

(2) y=150, w=15, so QSR is an isosceles triangle and now you have QS=2

when you have two length and its angle, you can get the third:

PQ^2=QS^2+PS^2-2QS*PS*sin30

sufficient
Manager
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Re: What is the perimeter of ∆PQS ?  [#permalink]

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11 Nov 2015, 00:19
Thanks for the responses! That's some quality discussion you guys have come up with.

So in conclusion, there 2 basic things we need to realize to answer this question:
1) As with every DS question, we dont have to 'find' the answer. We only need to prove that is can be found.
2) The 'sufficiency' to obtain a unique triangle-
Having two sides and 1 included angle
Having two angles and 1 included side
_________________
One Kudos for an everlasting piece of knowledge is not a bad deal at all...

------------------------------------------------------------------------------------------------------------------------
Twenty years from now you will be more disappointed by the things you didn't do than by the ones you did do. So throw off the bowlines. Sail away from the safe harbor. Catch the trade winds in your sails. Explore. Dream. Discover.
-Mark Twain
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Re: What is the perimeter of ∆PQS ?  [#permalink]

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25 Dec 2018, 09:05
Bunuel,

Could you help to solve this question.

Thanks,
Re: What is the perimeter of ∆PQS ?   [#permalink] 25 Dec 2018, 09:05
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