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Geometery - Triangles

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by RobertP » Sun Jan 24, 2016 11:31 am
In the figure bellow the triangles ABC and MNP are isosceles. AB || MN, BC || NP, the length of AC is 7 and the length of BY is 4. If the area if the white region is equal to the area of the grey region, what is the length of MP?



(A) 2 * 2^(1/2)
(B) 2 * 7^(1/2)
(C) ( 2 * 3^(1/2) ) / 3
(D) (7 * 2^(1/2) ) / 2
(E) (7 * 3^(1/2) ) / 3

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by MartyMurray » Sun Jan 24, 2016 2:49 pm
RobertP wrote:In the figure bellow the triangles ABC and MNP are isosceles. AB || MN, BC || NP, the length of AC is 7 and the length of BY is 4. If the area if the white region is equal to the area of the grey region, what is the length of MP?



(A) 2√2
(B) 2√7
(C) 2√3/3
(D) 7√2/2
(E) 7√3/3

Image
Since the sides of the two triangles are parallel and the two triangles share the same base, we know that the triangles are similar triangles.

The area of triangle ABC = bh/2 = (AC * BY)/2 = (7 * 4)/2 = 28/2 = 14.

If the area of the white region is equal to area of the grey region, then the area of each region is half of the area of triangle ABC.

Area of each region = 14/2 = 7

So the area of the white region, triangle MNP, is 7.

The proportions of the sides and heights of similar triangles are all the same.

b/h of ABC = 7/4

So b/h of MNP is also 7/4.

So area of MNP = 7 = (7x * 4x)/2

14 = 7x * 4x

14 = 28x²

1/2 = x²

x = √(1/2) = √2/2

MP = 7x = 7√2/2

The correct answer is D.
Last edited by MartyMurray on Sun Jan 24, 2016 3:02 pm, edited 1 time in total.
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by RobertP » Sun Jan 24, 2016 3:02 pm
Marty, thank you very much. I was having a really hard time with this question.

I understood everything and there were some concepts such as similar triangles that I did not know of.

However there was once passage that I did not understand:

1/2 = x²

x = √(1/2) = √2/2

Why is the square roote of (1/2) equal to the square root of 2/2?

Thank you.
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by MartyMurray » Sun Jan 24, 2016 3:12 pm
RobertP wrote:Marty, there was once passage that I did not understand:

1/2 = x²

x = √(1/2) = √2/2

Why is the square root of (1/2) equal to the square root of 2/2?
√(1/2) = √1/√2 = 1/√2

Usually people do not keep the root in the denominator.

Move the root to the numerator by multiplying top and bottom by √2.

√2/√2 * 1/√2 = √2/2
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by MartyMurray » Sun Jan 24, 2016 3:38 pm
Similar triangles show up a lot on the GMAT. So make sure that you understand how they can work.

For instance, in this diagram, triangles ACD and ABD are similar because they are both right triangles and also they share angle A.

Meanwhile ACD and CBD are also similar, because they are both right triangles and also they share angle C.

Image

You could read these posts to get some more insights.

https://www.beatthegmat.com/mba/2009/10/ ... art-1-of-7

https://magoosh.com/gmat/2013/gmat-math-similar-shapes/
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