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plz explain

by sana.noor » Thu Sep 26, 2013 4:27 am
1) A barn is enclosed in the shape of a 6-sided figure; all sides are the same length and all angles have the same measure. The distance from the center of the barn to the midpoint of one of the sides is 50√3 feet What is the area of the enclosed space?

i dont have its OA
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by theCodeToGMAT » Thu Sep 26, 2013 4:51 am
Assuming you meant the distance as √503.

According to me,

Area enclosed = 6 * area of equilateral triangle with altitude √503

√3/2 * side = √503 --> side = 2 * √(503/3)

Area = 6 * √3/4 * 4 * 503/3 = 1006√3
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by Resp007 » Thu Sep 26, 2013 5:03 am
hi,
Since every side and angles are same so we have a 6 equilateral triangles, with heights given for each. To find the total are we need base or one side of the triangle: (see fig)

So answer = 15*(root 3)*1000 sq units, assuming, ht = 503√ feet = 50* (root 3).

Does it make sense? :/
sana.noor wrote:1) A barn is enclosed in the shape of a 6-sided figure; all sides are the same length and all angles have the same measure. The distance from the center of the barn to the midpoint of one of the sides is 503√ feet What is the area of the enclosed space?

i dont have its OA
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by Brent@GMATPrepNow » Thu Sep 26, 2013 6:50 am
sana.noor wrote:1) A barn is enclosed in the shape of a 6-sided figure; all sides are the same length and all angles have the same measure. The distance from the center of the barn to the midpoint of one of the sides is 50√3 feet. What is the area of the enclosed space?
I'm assuming that the barn is in the center of this hexagon.

Since all sides are the same length, and all angles have the same measure, we have a "regular" hexagon.
Image

If we draw lines from the center to the 6 vertices, we get 6 equilateral triangles.
Image

The distance from the center of the barn to the midpoint of one of the sides is 50√3 feet. So, we get:
Image

Notice that we can now create a 30-60-90 right triangle.
Image

If we compare the red triangle to the "base" 30-60-90 triangle . . .
Image
. . . we can see that the red triangle is 50 times bigger than the "base" triangle [since 50√3 is 50 times greater than √3]

So, the other two sides of the red triangle must have lengths 50 and 100
Image

From here, we can see that each side of this equilateral triangle has length 100.
Image

IMPORTANT: For test day, be sure to memorize the formula for the area of an equilateral triangle. Area = √3(side)²/4 [yes, we can also use other triangle area formula here, but I wanted to point out another useful formula to know]

So, the area of this equilateral triangle = √3(100)²/4
= 2500√3

Of course, that's the area of just one of the 6 equilateral triangles.

So, the ENTIRE area of the hexagon = 6( 2500√3) = [spoiler]15,000√3[/spoiler]

Cheers,
Brent
Brent Hanneson - Creator of GMATPrepNow.com
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