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# Points M and P lie on square LNQR, and LM = LQ. What is the

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Points M and P lie on square LNQR, and LM = LQ. What is the  [#permalink]

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Updated on: 25 Oct 2013, 09:11
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Points M and P lie on square LNQR, and LM = PQ. What is the length of the line segment PQ?

(1) $$PR=\frac{4\sqrt{10}}{3}$$
(2) The ratio of the area of the unshaded region to the total area of the shaded region is 2 to 1.

Originally posted by Pmar2012 on 25 Oct 2013, 08:46.
Last edited by Bunuel on 25 Oct 2013, 09:11, edited 2 times in total.
Edited the question.
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Re: Points M and P lie on square LNQR, and LM = LQ. What is the  [#permalink]

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25 Oct 2013, 09:09
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Points M and P lie on square LNQR, and LM = PQ. What is the length of the line segment PQ?

(1) $$PR=\frac{4\sqrt{10}}{3}$$. We know two sides (PR and PQ) in right triangle PQR, thus we can find the third side PQ. Sufficient.

(2) The ratio of the area of the unshaded region to the total area of the shaded region is 2 to 1. Say LM = PQ = x, then the area of the shaded region is 2*(1/2*4*x)=4x. The area of unshaded region is 4*4-4x=16-4x. Thus we have that (unshaded)/(shaded)=(16-4x)/4x=2/1. We can find x. Sufficient.

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Re: Points M and P lie on square LNQR, and LM = LQ. What is the  [#permalink]

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11 Dec 2013, 09:53
Points M and P lie on square LNQR, and LM = PQ. What is the length of the line segment PQ?

(1) PR=\frac{4\sqrt{10}}{3}

This is pretty self explanatory. Sufficient.

(2) The ratio of the area of the unshaded region to the total area of the shaded region is 2 to 1.

Because we know that LM = PQ, and that the figure is a square, we know that both shaded triangles are equal to one another. We know that the total area of the square (including the shaded triangles) = 16. If the ratio of unshaded to shaded is 2:1 we can set up an equation.

x:16 = 2:3 Where x represents the unshaded portion relative to the entire square and 2:3 represents the ratio given to us.

x = 32/3. Now we can find the remaining area of the triangles (which are both equal to one another). Finally, we plug in the area and the one given base length (4) into the area of a triangle formula to get the missing length (i.e. PQ or ML.) Sufficient.

D
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Points M and P lie on square LNQR, and LM = LQ. What is the  [#permalink]

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21 Nov 2015, 19:19
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Easy solution:

A) Obviously sufficient because a^2+b^2=c^2

B) The white to gray is 2:1 so the gray is 1/3 of the total area. The area is 4*4. So the missing line segment is (4/3).

ANS: D
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Re: Points M and P lie on square LNQR, and LM = LQ. What is the  [#permalink]

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26 Nov 2015, 08:00
Forget conventional ways of solving math questions. In DS, Variable approach is the easiest and quickest way to find the answer without actually solving the problem. Remember equal number of variables and independent equations ensures a solution.

Points M and P lie on square LNQR, and LM = PQ. What is the length of the line segment PQ?

(1) PR=410 − − √ 3
(2) The ratio of the area of the unshaded region to the total area of the shaded region is 2 to 1.

We can know PQ if we know PR, so there is one variable (PR), and 2 equations are given by the conditions, so there is high chance (D) will be the answer.
For condition 1, from (4 sqrt 10/3)^2-4^2=PQ^2. PQ=4/3. This is sufficient
For condition 2, if the ratio of the area is 2:1, the height is the same, so NP:PQ=2:1, and PQ=4/3 so this is sufficient as well, and the answer becomes (D).

For cases where we need 1 more equation, such as original conditions with “1 variable”, or “2 variables and 1 equation”, or “3 variables and 2 equations”, we have 1 equation each in both 1) and 2). Therefore, there is 59 % chance that D is the answer, while A or B has 38% chance and C or E has 3% chance. Since D is most likely to be the answer using 1) and 2) separately according to DS definition. Obviously there may be cases where the answer is A, B, C or E.
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Re: Points M and P lie on square LNQR, and LM = LQ. What is the  [#permalink]

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22 Jul 2017, 19:17
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Pmar2012 wrote:
Attachment:
Untitled2.png
Points M and P lie on square LNQR, and LM = PQ. What is the length of the line segment PQ?

(1) $$PR=\frac{4\sqrt{10}}{3}$$
(2) The ratio of the area of the unshaded region to the total area of the shaded region is 2 to 1.

The goal is to find then length of PQ.

Statement 1) PR = 4sqrt(10)/3

By the Pythagorean theorem, we can determine that x^2 + 4^2 = 16/9*10

x^2 = 160/9-16

x^2 = 17.77-16

Sufficient.

Statement 2) The ratio of the area of the unshaded region to the total area of the shaded region is 2 to 1.

Let PQ = x

The area of the shaded region is 2(1/2*4*x) = 4x

The total area of the square is 4^2 = 16

(16-4x)/(4x) = 2/1

16-4x = 8 x

16 = 12x
4 = 3x
4/3 = x

Sufficient.
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Re: Points M and P lie on square LNQR, and LM = LQ. What is the  [#permalink]

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01 Jan 2019, 11:00
St1:- Apply pythagoras theorem as the lengths of two sides are given and one is missing.

St2:- Apply the formula for area of a triangle formula to find the length of the side.

Option D is the correct answer
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Re: Points M and P lie on square LNQR, and LM = LQ. What is the   [#permalink] 01 Jan 2019, 11:00
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