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# In the figure above triangles ABC and MNP are both isosceles

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In the figure above triangles ABC and MNP are both isosceles  [#permalink]

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Updated on: 14 Feb 2012, 08:26
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Difficulty:

65% (hard)

Question Stats:

69% (03:11) correct 31% (03:00) wrong based on 223 sessions

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In the figure above, triangles ABC and MNP are both isosceles. AB is parallel to MN, BC is parallel to NP, the length of AC is 7 and the length of BY is 4. If the area of the unshadd region is equal to the area of the shaded region, what is the length of MP?

A. $$2\sqrt{2}$$

B. $$2\sqrt{7}$$

C. $$\frac{2\sqrt{3}}{3}$$

D. $$\frac{7\sqrt{2}}{2}$$

E. $$\frac{7\sqrt{3}}{3}$$

Originally posted by rxs0005 on 14 Feb 2012, 07:59.
Last edited by Bunuel on 14 Feb 2012, 08:26, edited 1 time in total.
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14 Feb 2012, 08:14
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In the figure above, triangles ABC and MNP are both isosceles. AB is parallel to MN, BC is parallel to NP, the length of AC is 7 and the length of BY is 4. If the area of the unshadd region is equal to the area of the shaded region, what is the length of MP?

A. $$2\sqrt{2}$$

B. $$2\sqrt{7}$$

C. $$\frac{2\sqrt{3}}{3}$$

D. $$\frac{7\sqrt{2}}{2}$$

E. $$\frac{7\sqrt{3}}{3}$$

Since the area of unshaded region is equal to the are of shaded region, then the area of the big triangle is twice the area of the little triangle (unshaded region): $$\frac{AREA_{ABC}}{area_{MNP}}=\frac{2}{1}$$

Next, triangles ABC and MNP are similar. In two similar triangles, the ratio of their areas is the square of the ratio of their sides: $$\frac{AREA}{area}=\frac{S^2}{s^2}$$. Thus $$\frac{AREA_{ABC}}{area_{MNP}}=\frac{AC^2}{MP^2}$$ --> $$\frac{2}{1}=\frac{7^2}{MP^2}$$ --> $$MP^2=\frac{7^2}{2}$$ --> $$MP=\frac{7}{\sqrt{2}}=\frac{7\sqrt{2}}{2}$$.

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Re: In the figure above triangles ABC and MNP are both isosceles  [#permalink]

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14 Feb 2012, 08:35
Neat little rule Bunuel ! Kudos !
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31 Mar 2012, 17:02
Since the area of unshaded region is equal to the area of shaded region, then the area of the big triangle is twice the area of the little triangle (unshaded region): $$\frac{AREA_{ABC}}{area_{MNP}}=\frac{2}{1}$$

How did you get the above?
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31 Mar 2012, 17:07
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enigma123 wrote:
Since the area of unshaded region is equal to the area of shaded region, then the area of the big triangle is twice the area of the little triangle (unshaded region): $$\frac{AREA_{ABC}}{area_{MNP}}=\frac{2}{1}$$

How did you get the above?

Hope it's clear.
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Re: In the figure above triangles ABC and MNP are both isosceles  [#permalink]

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02 Sep 2019, 06:14
Hello from the GMAT Club BumpBot!

Thanks to another GMAT Club member, I have just discovered this valuable topic, yet it had no discussion for over a year. I am now bumping it up - doing my job. I think you may find it valuable (esp those replies with Kudos).

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Re: In the figure above triangles ABC and MNP are both isosceles   [#permalink] 02 Sep 2019, 06:14
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