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.
If we draw lines from the center to the 6 vertices, we get 6 equilateral triangles.
The distance from the center of the barn to the midpoint of one of the sides is
50√3 feet. So, we get:
Notice that we can now create a 30-60-90 right triangle.
If we compare the
red triangle to the "
base" 30-60-90 triangle . . .
. . . 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
From here, we can see that each side of this equilateral triangle has length
100.
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
