Equilateral Triangles - 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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Question

If the area of an equilateral is , given a base of , what is the height of the triangle?

Answer

We derive the height formula from the area of the triangle formula:

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

$h=\frac{2A}{b}$

$=\frac{(2*21)}{3}$

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Question

What is the height of an equilateral triangle with sidelength 20?

Answer

The area of an equilateral triangle with sidelength is

$A = \frac{s^{2}\sqrt{3}}{4} = \frac{20^{2}\sqrt{3}}{4} = \frac{400\sqrt{3}}{4} = 100\sqrt{3}$

Using this area for and 20 for in the general triangle formula, we can obtain :

$\frac{1}{2} bh = A$

$\frac{1}{2}\cdot 20 \cdot h =100\sqrt{3}$

$10 \cdot h =100\sqrt{3}$

$10 \cdot h \div 10 =100\sqrt{3}\div 10$

$h=10\sqrt{3}$

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Question

is an equilateral triangle, with a side length of . What is the height of the triangle?

Answer

We know the length of the side, therefore we can use the formula for the height in an equilateral triangle:

$h=s\frac{\sqrt{3}}{2}$ , where is the length of a side and the length of the height.

Therefore, the final answer is $\frac{3\sqrt{3}}{2}$ .

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Question

An equilateral triangle has a side length of . What is the height of the triangle?

Answer

The height of an upright equilateral triangle is the perpendicular distance from the center of its base to its top. We can imagine that this line cuts the equilateral triangle into two congruent right triangles whose height is half the length of the original base and whose hypotenuse is the original side length. In these two congruent triangles, their base, which is the height of the equilateral triangle, is the only unknown side length, so we can use the Pythagorean theorem to solve for it:

$(\frac{5}{2})^2+b^2=5^2$

$b^2=5^2-(\frac{5}{2})^2=\frac{75}{4}$

$b=\sqrt{\frac{75}{4}}=\frac{\sqrt{3*25}}{2}=\frac{5\sqrt{3}}{2}$

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Question

Given that an equilateral triangle has side lengths equal to , determine it's height in simplest form.

Answer

To solve, we must use pythagorean's theorem given that we know the hypotenuse is and one side length is $\left (\frac{1}{2} \textup{ of } 6 \right )$ . Therefore:

$c^2=a^2+b^2\Rightarrow 6^2=a^2+3^2\Rightarrow 27=a^2$

$a=\sqrt{27}=\sqrt{3^2*3}=3\sqrt3$

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Question

If an equilateral triangle has a perimeter of , what is the length of each side?

Answer

An equilateral triangle has three equal sides; therefore, to find the length of each side, divide the perimeter by :

$\frac{18}{3}= 6$

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Question

If the area of an equilateral is , given a height of , what is the base of the triangle?

Answer

We derive the equation of base of a triangle from the area of a triangle formula:

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

$b=\frac{2A}{h}$

$b=\frac{2*12}{5}$

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Question

The height of an equilateral triangle is . What is the length of side ?

Answer

Similarily, we can use the same formula for the height to find the length of the side of an equilateral triangle, which is given by

$s=\frac{2h}{\sqrt{3}}$ , where is the length of the height.

Therefore, the final answer is

$\frac{2\cdot 5}{\sqrt3}=\frac{10}{\sqrt{3}}$ .

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Question

Equilateral triangle is inscribed in a circle with radius , what is the length of a side of the triangle?

Answer

Since we are given the radius, we should be able to find the height of the equilateral triangle. Indeed, the center of the circle is at the intersections of the heights of the triangle, and is located $\frac{2}{3}$ away from the edge of a given height.

Therefore 5, the radius of the circle is $\frac{2}{3}$ of the height.

Therefore, the height must be $\frac{15}{2}$ .

From here, we can use the formula for the height of the equilateral triangle $h=s\frac{\sqrt{3}}{2}$ , where is the length of the height and is the length of a side of the equilateral triangle.

Therefore, $s=2\frac{h}{\sqrt{3}}$ , then $\frac{15}{\sqrt{3}}$ is the final answer.

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Question

The area of an equilateral triangle is . What is the perimeter of ?

Answer

The area is given, which will allow us to calculate the side of the triangle and hence we can also find the perimeter.

The area for an equilateral triangle is given by the formula

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

Therefore, $s=\sqrt{\frac{4a}{\sqrt{3}}}$ , where is the area.

Thus $s=\sqrt{\frac{28}{\sqrt{3}}}$ , and the perimeter of an equilateral triangle is three times the side, hence, the final answer is $3\sqrt{\frac{28}{\sqrt{3}}}$ .

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Question

Find the perimeter of an equilateral triangle whose side length is .

Answer

To find the perimeter, you must multiply the side length by . Thus,

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Question

Calculate the perimeter of an equilateral triangle whose side length is .

Answer

To solve, simply multiply the side length by . Thus,

$P=3\cdot 3=9$

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Question

A given equilateral triangle has a side length of . What is the perimeter of the triangle?

Answer

An equilateral triangle with a side length has a perimeter .

Given:

,

.

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