Equilateral Triangles - SSAT Upper Level Quantitative (Math)

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Question

An equilateral triangle is circumscribed about a circle of radius 16. Give the area of the triangle.

Answer

The circle and triangle referenced are below, along with a radius to one side and a segment to one vertex:

$\Delta ABO$ is a 30-60-90 triangle, so

$AB = OB \cdot \sqrt{3} = 16 \sqrt{3}$

$\overline{AB}$ is one-half of a side of the triangle, so the sidelength is $32\sqrt{3}$ . The area of the triangle is

$A = \frac{s^{2} \sqrt{3}}{4} = \frac{(32\sqrt{3})^{2} \sqrt{3}}{4} = \frac{1,024 \cdot 3 \cdot \sqrt{3}}{4}= 768 \sqrt{3}$

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Question

An equilateral triangle is inscribed inside a circle of radius . Give the area of the triangle.

Answer

The trick is to know that the circumscribed circle, or the circumcircle, has as its center the intersection of the three altitudes of the triangle, and that this center, or circumcenter, divides each altitude into two segments, one twice the length of the other - the longer one being a radius. Because of this, we can construct the following:

Each of the six smaller triangles is a 30-60-90 triangle, and all six are congruent.

We will find the area of $\Delta COY$ , and multiply it by 6.

By the 30-60-90 Theorem, $CY = OY \cdot \sqrt{3} = 21\sqrt{3}$ , so the area of $\Delta COY$ is

$A = \frac{1}{2}\cdot OY \cdot CY = \frac{1}{2} \cdot 21\cdot 21\sqrt{3} =\frac{ 441 \sqrt{3}}{2}$ .

Six times this - $6 \cdot \frac{ 441 \sqrt{3}}{2} = 1,323 \sqrt{3}$ - is the area of $\Delta ABC$ .

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Question

In the above diagram, $\Delta ABC$ is equilateral. Give its area.

Answer

The interior angles of an equilateral triangle all measure 60 degrees, so, by the 30-60-90 Theorem,

$BM = \frac{AM}{\sqrt{3}} = \frac{32}{\sqrt{3}}$

Also, is the midpoint of $\overline{BC}$ , so $BC = 2 \cdot BM = \frac{64}{\sqrt{3}}$ ; this is the base.

The area of this triangle is half the product of the base and the height :

$A= \frac{1}{2} \cdot 32 \cdot \frac{64}{\sqrt{3}}= \frac{1,024}{\sqrt{3}} = \frac{1,024\cdot \sqrt{3}}{\sqrt{3} \cdot \sqrt{3}}= \frac{1,024 \sqrt{3}}{3}$

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Question

The perimeter of an equilateral triangle is . Give its area.

Answer

An equilateral triangle with perimeter 54 has three congruent sides of length

$s = 51 \div 3 = 18$

The area of this triangle is

$A = \frac{s^{2}\sqrt{3}}{4} = \frac{18^{2}\sqrt{3}}{4} = \frac{324 \sqrt{3}}{4}= 81 \sqrt{3}$

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Question

The perimeter of an equilateral triangle is $30\sqrt{6}$ . Give its area.

Answer

An equilateral triangle with perimeter $30\sqrt{6}$ has three congruent sides of length

$s = 30\sqrt{6}\div 3 = 10\sqrt{6}$

The area of this triangle is

$A = \frac{s^{2}\sqrt{3}}{4} = \frac{ \left ( 10\sqrt{6} \right )^{2}\sqrt{3}}{4} = \frac{100 \cdot 6 \cdot \sqrt{3}}{4} = 150\sqrt{3}$

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Question

Hexagon is regular and has perimeter 72. $\Delta ACE$ is constructed. What is its area?

Answer

Since the perimeter of the (six-congruent-sided) regular hexagon is 72, each side has length one sixth this, or 12.

The figure described is given below, with a perpendicular segment drawn from to side $\overline{AC}$ :

Each angle of a regular hexagon measures $120^{\circ}$ . Therefore, $m \angle ABM = 60^{\circ }$ , and $\Delta ABM$ is a 30-60-90 triangle.

By the 30-60-90 Theorem,

$BM = \frac{1}{2}AB = \frac{1}{2} \cdot 12 = 6$

and

$AM = BM \sqrt{3} = 6 \sqrt{3}$

$AC = 2 \cdot AM = 2 \cdot 6 \sqrt{3} =12 \sqrt{3}$ .

$\Delta ACE$ is equilateral, and $12 \sqrt{3}$ is its sidelength, making its area

$A = \frac{s^{2}\sqrt{3}}{4} = \frac{\left (12 \sqrt{3} \right )^{2}\sqrt{3}}{4} = \frac{144 \cdot 3 \cdot \sqrt{3}}{4} =108 \sqrt{3}$

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Question

An equilateral triangle has side lengths of . What is the area of this triangle?

Answer

The area of an equilateral triangle can be found using this formula:

$\frac{\sqrt3}{4}(\text{side length})^2$

Using $\text{side length}=2x$ , we can find the area of the equilateral triangle.

$\text{Area}=\frac{\sqrt3}{4}4x^2=x^2\sqrt3$

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Question

An equilateral triangle has sides of length . Find the measure of the height for this triangle.

Answer

The height of an equilateral triangle will always cut the side it intersects in half. Now, this is just a matter of applying the Pythagorean Theorem.

$6^2=3^2+\text{height}^2$

$\text{height}^2=27$

$\text{height}=3\sqrt3$

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Question

The side of an equilateral triangle, in feet, is . What is the height of this triangle?

Answer

The triangle in question looks like this:

The height of a triangle will always cut one of the sides of an equilateral triangle in half. Now, to find the length of the height is just a matter of using the Pythagorean Theorem.

$9^2+\text{height}^2=18^2$

$\text{height}^2=243$

$\text{height}=9\sqrt3$

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Question

The length of a side of an equilateral triangle is feet. In feet, what is the height of this triangle?

Answer

The given equilateral should look similar:

Because the height of an equilateral triangle always cuts a side length in half, figuring out the height becomes a matter of applying the Pythagorean Theorem.

$\text{height}^2+7^2=14^2$

$\text{height}^2=147$

$\text{height}=7\sqrt3$

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Question

The length of a side of an equilateral triangle is centimeters. In centimeters, what is the length of the height of this triangle?

Answer

Draw out the equilateral triangle:

Since the height of an equilateral triangle will always cut one of the sides in half, find the height using the Pythagorean Theorem.

$\text{height}^2+5^2=10^2$

$\text{height}^2=75$

$\text{height}=5\sqrt3$

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Question

The length of a side of an equilateral triangle is . Find the length of the height of this triangle.

Answer

Draw out and label the triangle.

Even though you are given exponents for the lengths of this triangle, use the Pythagorean Theorem to solve it. The height of an equilateral triangle will always cut one of its bases in half.

$\text{height}^2+x^2=(2x)^2$

$\text{height}^2=3x^2$

$\text{height}=x\sqrt3$

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Question

Find the height of an equilateral triangle that has side lengths of .

Answer

Draw out and label the triangle.

Since the height of an equilateral triangle will always cut its base in half, use the Pythagorean Theorem to find the height.

$\text{height}^2+(6r)^2=(12r)^2$

$\text{height}^2=108r^2$

$\text{height}=6r\sqrt3$

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Question

Find the height of an equilateral triangle that has a side length of .

Answer

Draw and label this triangle:

Since the height of an equilateral triangle always cuts its base in half, use the Pythagorean Theorem to find the height of the triangle.

$\text{height}^2+(10z)^2=(20z)^2$

$\text{height}^2=300z^2$

$\text{height}=10z\sqrt3$

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Question

Find the height of an equilateral triangle, in yards, that has side lengths of yards.

Answer

Since the height of an equilateral triangle cuts the base in half, use the Pythagorean Theorem to find the height.

$\text{height}^2+14^2=28^2$

$\tet{height}^2=588$

$\text{height}=14\sqrt3$

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Question

An equilateral triangle has a perimeter of units. What is the length of each side?

Answer

Because an equilateral triangle has three sides that are the same length, divide the given perimeter by 3 to find the length of each side.

$2400\div3=800$

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Question

The perimeter of a equilateral triangle is . In meters, what is the length of a side of this triangle?

Answer

Since all three sides are equal in an equilateral triangle, we can just divide the perimeter by 3 to find a side length.

$99:m\div3:m=33:m$

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Question

The perimeter of an equilateral triangle is . What is the length of a side of this triangle?

Answer

Because all the sides in an equilateral triangle are equal, we can just divide the perimeter by 3 to find the length of a side.

$36x\div3=12x$

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Question

The perimeter of an equilateral triangle is 39 meters. In meters, how long is one side of the triangle?

Answer

An equilateral triangle has sides that are the same length. Then, to find the length of one side, you only need to divide the perimeter by 3.

$39\div3=13$

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Question

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

Answer

Since an equilateral triangle has sides that are all the same, divide the perimeter by to get the length of each side.

$120t\div3=40t$

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