Calculating the area of an equilateral triangle - GMAT Quantitative Reasoning

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Question

Three straight sticks are gathered of exactly equal length. They are placed end to end on the ground to form a triangle. If the area of the triangle they form is 1.732 square feet. What is the length in feet of each stick?

Answer

Let be the length of a side of an equilateral triangle. Then the formula for the area of an equilateral triangle with side is

$\frac{s^2\sqrt{3}}{4}$

So solving $1.732 = \frac{s^2\sqrt{3}}{4}$

we get .

Alternative Solution:

Without knowing this formula you can still use the Pythagorean Theorem to solve this. By drawing the height of the triangle, you split the triangle into 2 right triangles of equal size. The sides are the height, $\frac{s}{2}$ and . Letting stand for the unknown height, we solve

$(\frac{s}{2})^2 + h^2= s^2$ solving for we get

$h=\sqrt(s^2-(\frac{s}{2})^2) = \sqrt(s^2-\frac{s^2}{4}) = \sqrt(\frac{3s^2}{4}) = \frac{s\sqrt(3)}{2}$

The area for any triangle is the base times the height divided by 2. So

$1.732=\frac{sh}{2} = \frac{s(s\sqrt(3))}{2*2} = \frac{s^2\sqrt(3)}{4}$ or .

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Question

If an equilateral triangle has a side length of and a height of , what is the area of the given triangle?

Answer

To find the area of a traingle, we need the height and base lengths. Plug the given values into the following formula:

$A= \frac{bh}{2}$

$= \frac{(3*2)}2 {}$

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Question

Triangle is an equilateral triangle with side length . What is the area of the triangle?

Answer

The area of an equilateral triangle is given by the following formula:

$\frac{s^{2}\sqrt{3}}{4}$ , where is the length of a side.

Since we know the length of the side, we can simply plug it in the formula and we have $\frac{4\sqrt{3}}{4}$ or $\sqrt{3}$ , which is the final answer.

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Question

is an equilateral triangle inscribed in a cirlce with radius . What is the area of the triangle ?

Answer

Since we are given a radius for the circle, we should be able to find the length of the height of the equilateral triangle, indeed, the center of the circle is $\frac{2}{3}$ of the length of the height from any vertex.

Therefore, the height is $3=\frac{2}{3}\cdot h$ where is the length of the height of the triangle. Therefore $h=\frac{9}{2}$ .

We can now plug in this value in the formula of the height of an equilateral triangle $h=s\frac{\sqrt{3}}{2}$ , where is the length of the side of the triangle.

Therefore, $s=\frac{9}{\sqrt{3}}$ or $3\sqrt{3}$ .

Now we should plug in this value into the formula for the area of an equilateral triangle $a=s^{2}\frac{\sqrt{3}}{4}$ where is the value of the area of the equilateral triangle. Therefore $a= 27\frac{\sqrt{3}}{4}$ , which is our final answer.

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Question

A given right triangle has a base length and a height . What is the area of the triangle?

Answer

For a given right triangle with a side length and a height , the area is

$A=\frac{1}{2}bh$ . Plugging in the values provided:

$A=\frac{1}{2}(5)(4)$

$A=\frac{1}{2}(20)$

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Question

A given equilateral triangle has a side length and a height . What is the area of the triangle?

Answer

For a given equilateral triangle with a side length and a height , the area is

$A=\frac{1}{2}bh$ . Plugging in the values provided:

$A=\frac{1}{2}(4)(7)$

$A=\frac{1}{2}(28)$

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Question

A given right triangle has a base of length and a height . What is the area of the triangle?

Answer

For a given right triangle with a side length and a height , the area is

$A=\frac{1}{2}bh$ . Plugging in the values provided:

$A=\frac{1}{2}(9)(8)$

$A=\frac{1}{2}(72)$

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