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In the diagram, JKLM is a square, and P is the midpoint of KL. Is JQM

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In the diagram, JKLM is a square, and P is the midpoint of KL. Is JQM  [#permalink]

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New post 18 Mar 2015, 04:08
2
6
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A
B
C
D
E

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  55% (hard)

Question Stats:

64% (02:10) correct 36% (02:19) wrong based on 222 sessions

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In the diagram, JKLM is a square, and P is the midpoint of KL. Is JQM an equilateral triangle?

(1) Angle KPQ = 90°
(2) Angle JQP = 150°

Kudos for a correct solution.

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In the diagram, JKLM is a square, and P is the midpoint of KL. Is JQM  [#permalink]

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New post 18 Mar 2015, 15:47
Question: is JQM an equilateral triangle? OR is JQ=QM=JM? OR are angles MJQ, JQM, and QMJ equal to 60?

(1) Not sufficient. That just tells us that JKPQ is a right trapezoid with to angles that are equal to 90. However, the other two angles of JKPQ which would help us find the angles of the triangle are still unknown.
(2) Sufficient. Knowing that JQP=150 and that P is a midpoint of KL, we can find the angle of JQM: 360- 2*150 =60. Knowing that P is a midpoint indicates that JQ=QM, thus their angles are equal. (180-60)/2=60. So all the angles of the triangle JQM are equal to 60, hence it is equilateral.

Answer B.
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In the diagram, JKLM is a square, and P is the midpoint of KL. Is JQM  [#permalink]

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New post Updated on: 19 Mar 2015, 04:34
1
I think C

(1) Angle KPQ = 90° => There are many points which satisfy this conditions, i mean it lies on the lines which is the perpendicular with JM and pass on the midpont of JM => Insufficient
(2) Angle JQP = 150° => Also, there are many points satisfy this conditions, i mean it lies on the circular => Insufficient

Combine (1) (2) => Q is the intersect between the lines in (1) and the circLE in (2)

=> answer: C

Originally posted by camlan1990 on 18 Mar 2015, 19:06.
Last edited by camlan1990 on 19 Mar 2015, 04:34, edited 2 times in total.
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In the diagram, JKLM is a square, and P is the midpoint of KL. Is JQM  [#permalink]

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New post 19 Mar 2015, 03:29
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Bunuel wrote:
Attachment:
The attachment ghdmpp_img5.png is no longer available
In the diagram, JKLM is a square, and P is the midpoint of KL. Is JQM an equilateral triangle?

(1) Angle KPQ = 90°
(2) Angle JQP = 150°

Kudos for a correct solution.



attached image for my solution to this problem .

minor correction , Corrected the angle names..
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Re: In the diagram, JKLM is a square, and P is the midpoint of KL. Is JQM  [#permalink]

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New post 19 Mar 2015, 07:55
2
(1) INSUFFICIENT- only with KPQ= 90 and no information about length of PQ. if PQ increases then JQM increases. so we need additional information.
(2) INSUFFICIENT- PQJ= 150. but no information whether JK || PQ or not. So with P being the mid point and KPQ \(\neq{90}\), PQJ can be 150. We need additional information.

Considering both

KPQ= 90 & PQJ= 150

=>KJP= 30=>QJM=60. same way it can be shown that JMQ= 60. ultimately JQM=60.
Hence JQM= equilateral.

Ans- C.
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Re: In the diagram, JKLM is a square, and P is the midpoint of KL. Is JQM  [#permalink]

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New post 23 Mar 2015, 05:13
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Bunuel wrote:
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In the diagram, JKLM is a square, and P is the midpoint of KL. Is JQM an equilateral triangle?

(1) Angle KPQ = 90°
(2) Angle JQP = 150°

Kudos for a correct solution.


MAGOOSH OFFICIAL SOLUTION:

First of all, clearly if we knew that that JQM is equilateral, then we could know that ÐKPQ = 90° and that ÐJQP = 150°. That much is clear. An equilateral triangle is consistent with the statements, but the question is, does either statement or do both of them necessitate that JQM be equilateral?

Statement #1: Angle KPQ = 90°

This statement guarantees that Q is directly below P, so that PQ lies on the vertical midline of the square. This guarantees that JQM is isosceles, but Q could be at any height.
Attachment:
ghdmpp_img17.png
ghdmpp_img17.png [ 4.18 KiB | Viewed 4280 times ]


This statement guarantees that JQM is isosceles. It may or may not be equilateral. This statement, alone and by itself, is not sufficient.

Statement #2: Angle JQP = 150°

Now, that obtuse angle is fixed, but we don’t know whether Q is still on that vertical midline. This could be the case in which JQM is equilateral, or it could be entirely asymmetrical.
Attachment:
ghdmpp_img20.png
ghdmpp_img20.png [ 5.71 KiB | Viewed 4276 times ]


With the restriction of this statement, it could be true that triangle JQM is equilateral, but as in these diagram, JQM might be completely asymmetrical. We can’t give a definitive answer. This statement, alone and by itself, is not sufficient.

Combined statements: Angle KPQ = 90° & angle JQP = 150°

The first statement tells us that PQ is on the midline, and that JQM must be isosceles. The entire diagram is symmetrical over this vertical midline. This means, if ÐJQP = 150°, then ÐMQP = 150° as well, and because the angles around point Q must add up to 360°, this means that ÐJQM must equal 60°. Well, any isosceles with one 60° angle must be equilateral. We absolute know that triangle JQM must be isosceles now. This is definitive. The statements together are sufficient.

Answer = (C)
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Re: In the diagram, JKLM is a square, and P is the midpoint of KL. Is JQM  [#permalink]

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New post 24 Dec 2016, 06:37
Bunuel wrote:
Attachment:
ghdmpp_img5.png
In the diagram, JKLM is a square, and P is the midpoint of KL. Is JQM an equilateral triangle?

(1) Angle KPQ = 90°
(2) Angle JQP = 150°

Kudos for a correct solution.


Original Statement says P is midpoint of KL, so we can say some mirror image of P on JM must be a midpoint too.
To prove it an equilateral triangle, we need to find atleast one angle or 3 sides.

1) Ang KPQ=90 => PQ, if extended, will meet the JM at the mirrored midpoint, bisecting the base of the triangle in 2. This is a property of isosceles & equilateral triangles. --> NOT SUFFICIENT
2) Ang JQP=150 => No information regarding angles of triangle or sides --> NOT SUFFICIENT

1) & 2)
When PQ is extended, it forms 180degs (say PQS) => Ang JQS=AngPQS-AngJQP=30; Similarly, AngMQS=30 => Ang JQM=60

Hence C :)
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Re: In the diagram, JKLM is a square, and P is the midpoint of KL. Is JQM  [#permalink]

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New post 07 Aug 2018, 19:04
Fairly straightforward.

KPQJ is a quadrilateral hence the sum of interior angles must equal 360. Likewise, the triangle's interior angles must equal 180.

A says KPQ is 90. Clearly this is insufficient by itself as it means nothing
B says JPQ is 150 which means half of angles Q must be 30 as it forms a straight line with P; hence, Q is 60. Just because Q is 60 does not mean the remaining 2 interior angles of the triangle are 60. Insufficient.

Combining both:
We know JPQ is 150 and KPQ is 90. We also know all interior angles of a square connecting 2 sides is 90. Hence, angles J and K are 90 each. Regardless, K + P + Q + J = 360. We know from above K = 90, P = 90, PQJ = 150; hence, J outside of triangle is 360 - 330 = 30. Since whole of J is 90, 90-30 = 60. 2 angles = 60 --> third angle is 60.
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Re: In the diagram, JKLM is a square, and P is the midpoint of KL. Is JQM &nbs [#permalink] 07 Aug 2018, 19:04
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