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A hexagonal playing field is composed of six

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A hexagonal playing field is composed of six

by BTGmoderatorDC » Sun Jan 14, 2018 3:53 pm
A hexagonal playing field is composed of six equilateral triangles, each the same size. If one player controls each triangle, how much area is controlled by all six players?

(1) One edge of one triangle equals 15 feet.
(2) The perimeter of one triangular region equals 45 feet.

What's the best way to determine whether statement 1 is sufficient?

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Source: — Data Sufficiency |

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by ErikaPrepScholar » Mon Jan 15, 2018 7:35 am
We know that each triangle is equilateral. This means that each side is the same length. If we happen to know that we can solve for the area of an equilateral triangle using
$$A=\frac{\sqrt{3}}{4}a^2$$
we should realize that if we know one side of a triangle, we know the area of a triangle and can solve.

However, even if we don't know that, we can envision dividing our equilateral triangle in half, giving two right triangles with a base of a/2 and a hypotenuse of a. Using the pythagorean theorem, we can then solve for the height by plugging in
$$\frac{a}{2}^2+h^2=a^2$$
Once we found the height, we could then solve for the area of any triangle using
$$A=\frac{1}{2}bh$$
This would give us the equilateral triangle equation above. HOWEVER, we don't need to go through and do this - since this is DS, all we need to do is know that if we have a (one side of the triangle), we could find the area of a triangle and solve the problem.

Statement 1

This gives us one side of the triangle, which is exactly what we need. Sufficient.

Statement 2

In an equilateral triangle, all three sides are equal, so P = 3a, where P is perimeter. If the perimeter is 45, then 45 = 3a, and a = 15. Again, we have one side of the triangle. Sufficient.
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