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ISEE Upper Level Math : Triangles

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Example Question #3 : Equilateral Triangles

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

Possible Answers:

$3\text{in}$

$8\text{in}$

$4\text{in}$

$7\text{in}$

$6\text{in}$

Correct answer:

$6\text{in}$

Explanation:

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

P = 3a

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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Example Question #4 : Equilateral Triangles

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

Possible Answers:

$12\text{in}$

$9\text{in}$

$16\text{in}$

$13\text{in}$

$14\text{in}$

Correct answer:

$13\text{in}$

Explanation:

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

P = 3a

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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Example Question #5 : Equilateral Triangles

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

Possible Answers:

$6\sqrt{3}\ in^2$

$9\ in^2$

$9\sqrt{2}\ in^2$

$6\ in^2$

$9\sqrt{3}\ in^2$

Correct answer:

$9\sqrt{3}\ in^2$

Explanation:

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

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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Example Question #1 : Equilateral Triangles

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

Possible Answers:

$25\sqrt{3}$

150

$25\sqrt{6}$

$50\sqrt{3}$

Correct answer:

Explanation:

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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Example Question #7 : Equilateral Triangles

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

Possible Answers:

$27\sqrt{3}$

$36\sqrt{3}$

The correct answer is not among the other four choices.

Correct answer:

$27\sqrt{3}$

Explanation:

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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Example Question #8 : Equilateral Triangles

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

Possible Answers:

$\frac{n^{2} }{12}$

$\frac{n^{2} \sqrt{3}}{18}$

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

$\frac{n^{2} }{6}$

$\frac{n^{2} \sqrt{3}}{12}$

Correct answer:

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

Explanation:

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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Example Question #9 : Equilateral Triangles

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

Possible Answers:

$48\sqrt{2}$

$48\sqrt{3}$

$64\sqrt{3}$

Correct answer:

$48\sqrt{3}$

Explanation:

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:

Circumcircle

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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Example Question #1 : Equilateral Triangles

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

Possible Answers:

$24\sqrt{3}$

$36\sqrt{3}$

$12\sqrt{3}$

Correct answer:

$36\sqrt{3}$

Explanation:

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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Example Question #1 : How To Find The Area Of An Equilateral Triangle

Equilateral

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

Possible Answers:

$1452 \sqrt{3}$

2,178

$726 \sqrt{3}$

$726 \sqrt{2}$

1452

Correct answer:

$1452 \sqrt{3}$

Explanation:

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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Example Question #2 : How To Find The Area Of An Equilateral Triangle

Icecreamcone

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

Possible Answers:

$24\sqrt{3}$

$48\sqrt{3}$

$36 \sqrt{3}$

$12 \sqrt{3}$

Correct answer:

$48\sqrt{3}$

Explanation:

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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ISEE Upper Level Math : Triangles

Study concepts, example questions & explanations for ISEE Upper Level Math

All ISEE Upper Level Math Resources

Example Questions

Example Question #3 : Equilateral Triangles

Example Question #4 : Equilateral Triangles

Example Question #5 : Equilateral Triangles

Example Question #1 : Equilateral Triangles

Example Question #7 : Equilateral Triangles

Example Question #8 : Equilateral Triangles

Example Question #9 : Equilateral Triangles

Example Question #1 : Equilateral Triangles

Example Question #1 : How To Find The Area Of An Equilateral Triangle

Example Question #2 : How To Find The Area Of An Equilateral Triangle

All ISEE Upper Level Math Resources

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