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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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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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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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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]
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]
Hi Bunuel, please could you explain the highlighted part again?

Bunuel wrote:

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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Statement 1: The length of the sides PR and QR are given. Since PQR is a right triangle , we can find the third side PQ. Hence Statement 1 is sufficient.
Statement 2:
The ratio of the area of the unshaded region to the total area of the shaded region is 2 to 1.
Since we know the total area of the square ( 4 * 4) , we can find the total area of the shaded region by using the ratios.

Total area of shaded region = 1/3 *( Area of the square)

Lets consider 2 triangles , MLR and PQR

Area of triangle = ½ * base* height
Its given that LM = PQ,
So the base and height of triangle MLR and PQR are the same.
Thus, Area of triangle MLR = Area of triangle PQR

Area of triangle PQR = ½ (Area of total shaded region)
Once you know the area of PQR, we can easily find the length of PQ.
Area of PQR = ½ * QR * PQ =½ * 4 * PQ
Hence Statement 2 is sufficient.

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