Equilateral Triangles - ISEE Upper Level: Quantitative Reasoning

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Question

$\Delta ABC$ is an equilateral triangle. Points are the midpoints of $\overline{AB},\overline{BC},\overline{AC}$ , respectively. $\Delta DEF$ is constructed.

Which is the greater quantity?

(a) The area of $\Delta ABC$

(b) Twice the area of $\Delta DEF$

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Answer

If segments are constructed in which the endpoints form the midpoints of the sides of a triangle, then four triangles, congruent to each other and similar to the larger triangle, are formed. Therefore, one of these triangles - specifically, $\Delta DEF$ - would have one-fourth the area of $\Delta ABC$ . This means $\Delta ABC$ has more than twice the area of $\Delta DEF$ .

Note that the fact that the triangle is equilateral is irrelevant.

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Question

Which of the following could be the three sidelengths of an equilateral triangle?

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Answer

By definition, an equilateral triangle has three sides of equal length. We can identify the equilateral triangle by converting the given sidelengths to the same units and comparing them.

We can eliminate the following by showing that at least two sidelengths differ.

$2 \textrm{ yd, } 84 \textrm{ in, }7\textrm{ ft}$

2 yards = $2 \times 3 = 6$ feet.

Two sides have lengths 6 feet and 7 feet, so we can eliminate this choice.

$1\frac{1}{2} \textrm{ yd, } 50 \textrm{ in, }4\textrm{ ft}$

4 feet = $4 \times 12 = 48$ inches

Two sides have lengths 48 inches and 50 inches, so we can eliminate this choice.

$1\frac{1}{3} \textrm{ yd, } 48 \textrm{ in, }5\textrm{ ft}$

5 feet = $5 \times 12 = 60$ inches

Two sides have lengths 48 inches and 60 inches, so we can eliminate this choice.

$1\frac{2}{3} \textrm{ yd, } 48 \textrm{ in, }4\textrm{ ft}$

$1\frac{2}{3}$ yards = $1\frac{2}{3} \times 3 = 5$ feet

Two sides have lengths 4 feet and 5 feet, so we can eliminate this choice.

$\frac{2}{3} \textrm{ yd, } 24 \textrm{ in, }2\textrm{ ft}$

$\frac{2}{3}$ yards = $\frac{2}{3} \times3 = 2$ feet = $2 \times 12 = 24$ inches

All three sides have the same length, making this the triangle equilateral. This choice is correct.

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Question

The length of one side of an equilateral triangle is 6 inches. Give the area of the triangle.

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Answer

$Area=\frac{1}{2}absinC$ ,

where and are the lengths of two sides of the triangle and is the angle measure.

In an equilateral triangle, all of the sides have the same length, and all three angles are always $60^{\circ}$ .

$Area=\frac{1}{2}absinC=\frac{1}{2}\times 6\times 6\times sin60^{\circ}$

$sin60^{\circ}=\frac{\sqrt{3}}{2}\Rightarrow Area=\frac{1}{2}\times 6\times 6\times \frac{\sqrt{3}}{2}=9\sqrt{3}\ in^2$

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Question

The perimeter of an equilateral triangle is $30\sqrt[4]{3}$ . Give its area.

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Answer

An equilateral triangle with perimeter $30\sqrt[4]{3}$ has three congruent sides of length

$s = 30\sqrt[4]{3} \div 3 = 10\sqrt[4]{3}$

The area of this triangle is

$A = \frac{s^{2}\sqrt{3}}{4} = \frac{ \left ( 10\sqrt[4]{3} \right )^{2}\sqrt{3}}{4} = \frac{100 \sqrt[4]{9} \cdot \sqrt{3}}{4}$

$\sqrt[4]{9} = \sqrt[4]{3^{2}} = 3^{2/4} = 3^{1/2} = \sqrt{3}$ , so

$A = \frac{100 \sqrt{3} \cdot \sqrt{3}}{4} = \frac{100 \cdot 3 }{4}= 75$

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Question

The perimeter of an equilateral triangle is $18\sqrt{3}$ . Give its area.

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Answer

An equilateral triangle with perimeter $18\sqrt{3}$ has three congruent sides of length

$s = 18\sqrt{3} \div 3 = 6\sqrt{3}$

The area of this triangle is

$A = \frac{s^{2}\sqrt{3}}{4} = \frac{ \left ( 6\sqrt{3} \right )^{2}\sqrt{3}}{4} = \frac{36 \cdot 3 \cdot \sqrt{3}}{4} = 27\sqrt{3}$

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Question

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

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Answer

An equilateral triangle with perimeter has three congruent sides of length $\frac{n}{3}$ . Substitute this for in the following area formula:

$A =s^{2} \cdot \frac{\sqrt{3}}{4}$

$A = \left ( \frac{n}{3} \right )^{2} \cdot \frac{\sqrt{3}}{4}$

$A = \frac{n^{2} }{9} \cdot \frac{\sqrt{3}}{4}$

$A = \frac{n^{2} \sqrt{3}}{36}$

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Question

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

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Answer

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

$BM = \frac{AM}{\sqrt{3}} = \frac{66}{\sqrt{3}} = \frac{66\cdot \sqrt{3}}{\sqrt{3}\cdot \sqrt{3}} = \frac{66 \sqrt{3}}{3} =22 \sqrt{3}$

Also, is the midpoint of $\overline{BC}$ , so $BC = 2 \cdot BM = 44 \sqrt{3}$ ; this is the base.

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

$A= \frac{1}{2} \cdot 66 \cdot 44 \sqrt{3}= 1,452 \sqrt{3}$

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Question

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

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Answer

An equilateral triangle with perimeter 36 has three congruent sides of length

$s = 36 \div 3 = 12$

The area of this triangle is

$A = \frac{s^{2}\sqrt{3}}{4} = \frac{12^{2}\sqrt{3}}{4} = \frac{144\sqrt{3}}{4} = 36\sqrt{3}$

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Question

An equilateral triangle is inscribed inside a circle of radius 8. Give its area.

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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, $CY = OY \cdot \sqrt{3} = 4\sqrt{3}$ , so the area of $\Delta COY$ is

$A = \frac{1}{2}\cdot OY \cdot CY = \frac{1}{2} \cdot 4\cdot 4\sqrt{3} = 8 \sqrt{3}$ .

Six times this - $6 \cdot 8 \sqrt{3} = 48 \sqrt{3}$ - is the area of $\Delta ABC$ .

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Question

Refer to the above figure. The shaded region is a semicircle with area $24 \pi$ . Give the area of $\bigtriangleup ABC$ .

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Answer

Given the radius of a semicircle, its area can be calculated using the formula

$A = \frac{1}{2} \pi r ^{2}$ .

Substituting $A = 24 \pi$ :

$\frac{1}{2} \pi r ^{2} = 24 \pi$

$\frac{2}{\pi} \cdot \frac{1}{2} \pi r ^{2} = \frac{2}{\pi} \cdot 24 \pi$

$r ^{2} = 48$

$r = \sqrt{48} = \sqrt{16} \cdot \sqrt{3} = 4\sqrt{3}$

The diameter of this semicircle is twice this, which is $d = 2 \cdot 4\sqrt{3} = 8 \sqrt{3}$ ; this is also the length of $\overline{BC}$ .

$\bigtriangleup ABC$ has two angles of degree measure 60; its third angle must also have measure 60, making $\bigtriangleup ABC$ an equilateral triangle with sidelength $s = 8 \sqrt{3}$ . Substitute this in the area formula:

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

$= \frac{( 8 \sqrt{3})^{2}\sqrt{3}}{4}$

$= \frac{8^{2} \cdot ( \sqrt{3})^{2}\sqrt{3}}{4}$

$= \frac{64 \cdot 3 \cdot \sqrt{3}}{4}$

$= \frac{192 \sqrt{3}}{4}$

$= 48\sqrt{3}$

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Question

The length of one side of an equilateral triangle is 4 centimeters. Give the height (altitude) of the triangle.

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Answer

$Area=\frac{1}{2}absinC$ ,

where and are the lengths of two sides and is the corresponding angle.

In an equilateral triangle, all of the sides have the same length, and all three angles are $60^{\circ}$ .

$Area=\frac{1}{2}absinC=\frac{1}{2}\times 4\times 4\times sin60^{\circ}$

$sin60^{\circ}=\frac{\sqrt{3}}{2}\Rightarrow Area=\frac{1}{2}\times 4\times 4\times \frac{\sqrt{3}}{2}=4\sqrt{3}\ cm^2$

Now plug this area into the alternate formula for the area of the triangle and solve for the height:

$Area =\frac{bh}2{}\Rightarrow 4\sqrt{3}=\frac{4\times h}{2}\Rightarrow h=2\sqrt{3}\ cm$

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Question

The height of the equilateral triangle below is 5 centimeters. Give the perimeter of the triangle.

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Answer

In an equilateral triangle all of the sides have the same length, so all we need to do is find the length of one side.

We know that, in an equilateral triangle, all three angles measure $60^{\circ}$ .

$sin\angle A=\frac{h}{AB}=\frac{h}{c}$

$sin\angle A=\frac{h}{c}\Rightarrow sin60^{\circ}=\frac{4}{c}\Rightarrow \frac{\sqrt{3}}{2}=\frac{4}{c}\Rightarrow c=\frac{8}{\sqrt{3}}=\frac{8\sqrt{3}}{3} \ cm$

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Question

For an equilateral triangle, Side A measures and Side B measures . What is the length of Side A?

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Answer

First you need to recognize that for an equilateral triangle, all 3 sides have equal lengths.

This means you can set the two values for Side A and Side B equal to one another, since they measure the same length, to solve for .

You now know that , but this is not your answer. The question asked for the length of Side A, so you need to plug 3 into that equation.

So the length of Side A (and Side B for that matter) is 8.

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Question

One angle of an equilateral triangle is 60 degrees. One side of that triangle is 12 centimeters. What are the measures of the two other angles and two other sides?

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Answer

An equilateral triangle is one in which all three sides are congruent. It also has the property that all three interior angles are equal. In other words, all three angles of an equilateral triangle are always 60°. Since all sides are congruent, the other two sides both measure 12 centimeters.

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Question

If an equilateral triangle has a perimeter of 18in, what is the length of one side?

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Answer

To find the perimeter of an equilateral triangle, we will use the following formula:

where a is the length of any side of the triangle. Because an equilateral triangle has 3 equal sides, we can use any side in the formula. To find the length of one side of the triangle, we will solve for a.

Now, we know the perimeter of the equilateral triangle is 18in. So, we will substitute.

$18\text{in} = 3a$

$\frac{18\text{in}}{3} = \frac{3a}{3}$

$6\text{in} = a$

$a=6\text{in}$

Therefore, the length of one side of the equilateral triangle is 6in.

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Question

An equilateral triangle has a perimeter of 39in. Find the length of one side.

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Answer

An equilateral triangle has 3 equal sides. So, to find the length of one side, we will use what we know. We know the perimeter of the equilateral triangle is 39in. So, we will look at the formula for perimeter. We get

where a is the length of one side of the triangle. So, to find the length of one side, we will solve for a. Now, as stated before, we know the perimeter of the triangle is 39in. So, we will substitute and solve for a. We get

$39\text{in} = 3a$

$\frac{39\text{in}}{3} = \frac{3a}{3}$

$13\text{in} = a$

$a = 13\text{in}$

Therefore, the length of one side of the equilateral triangle is 13in.

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Question

Refer to the above figure. The shaded region is a semicircle with area $16 \pi$ . Give the perimeter of $\bigtriangleup ABC$ .

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Answer

Given the radius of a semicircle, its area can be calculated using the formula

$A = \frac{1}{2} \pi r ^{2}$ .

Substituting $A = 16 \pi$ :

$\frac{1}{2} \pi r ^{2} = 16 \pi$

$\frac{2}{\pi} \cdot \frac{1}{2} \pi r ^{2} = \frac{2}{\pi} \cdot 16 \pi$

$r ^{2} = 32$

$r = \sqrt{32} = \sqrt{16} \cdot \sqrt{2} = 4\sqrt{2}$

The diameter of this semicircle is twice this, which is $d = 2 \cdot 4\sqrt{2} = 8 \sqrt{2}$ ; this is also the length of $\overline{BC}$ .

$\bigtriangleup ABC$ has two angles of degree measure 60; its third angle must also have measure 60, making $\bigtriangleup ABC$ an equilateral triangle with sidelength $s = 8 \sqrt{2}$ . Its perimeter is three times this, or $3 \cdot 8 \sqrt{2} = 24 \sqrt{2}$

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Question

Find the perimeter of an equilateral triangle with a base of length 8cm.

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Answer

To find the perimeter of a triangle, we will use the following formula:

where a, b, and c are the lengths of the sides of the triangle

Now, we know the base of the triangle is 8cm. Because it is an equilateral triangle, all sides are equal. Therefore, all sides are 8cm.

Knowing this, we can substitute into the formula. We get

$P = 8\text{cm} + 8\text{cm} + 8\text{cm}$

$P = 24\text{cm}$

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Question

Find the perimeter of an equilateral triangle with a base of 21in.

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Answer

To find the perimeter of a triangle, we will use the following formula:

where a, b, and c are the lengths of the sides of the triangle.

Now, we know the base of the triangle has a length of 21in. Because it is an equilateral triangle, all lengths are the same. Therefore, all lengths are 21in.

Knowing this, we can substitute into the formula. We get

$P = 21\text{in} +21\text{in} +21\text{in}$

$P = 63\text{in}$

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Question

Find the perimeter of an equilateral triangle with a base of 23in.

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Answer

To find the perimeter of a triangle, we will use the following formula:

where a, b, and c are the lengths of the sides of the triangle.

Now, we know the base of the triangle has a length of 23in. Because it is an equilateral triangle, all lengths are the same. Therefore, all lengths are 23in.

Knowing this, we can substitute into the formula. We get

$P = 23\text{in} +23\text{in} +23\text{in}$

$P = 69\text{in}$

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