Equilateral Triangles

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ISEE Upper Level Quantitative Reasoning › Equilateral Triangles

Questions 1 - 10

Icecreamcone

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

$24 \sqrt{2}$

$24 \sqrt{3}$

$12 \sqrt{2}$

$12 \sqrt{3}$

Explanation

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

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

Substituting $A = 16 \pi$ :

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

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

$r ^{2} = 32$

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

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 $3 \cdot 8 \sqrt{2} = 24 \sqrt{2}$

Icecreamcone

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

$24 \sqrt{2}$

$24 \sqrt{3}$

$12 \sqrt{2}$

$12 \sqrt{3}$

Explanation

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

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

Substituting $A = 16 \pi$ :

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

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

$r ^{2} = 32$

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

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 $\overline{BC}$ .

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

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

$9\ in^2$

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

$6\ in^2$

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

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 $60^{\circ}$ .

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

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

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

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

$9\ in^2$

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

$6\ in^2$

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

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 $60^{\circ}$ .

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

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

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

$13\text{in}$

$16\text{in}$

$12\text{in}$

$9\text{in}$

$14\text{in}$

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

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.

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

$13\text{in}$

$16\text{in}$

$12\text{in}$

$9\text{in}$

$14\text{in}$

Explanation

$39\text{in} = 3a$

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

$13\text{in} = a$

$a = 13\text{in}$

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

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

$69\text{in}$

$46\text{in}$

$58\text{in}$

$264\text{in}$

$128\text{in}$

Explanation

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

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

$P = 69\text{in}$

Find the perimeter of an equilateral triangle with a base of 22in.

$66\text{in}$

$242\text{in}$

$44\text{in}$

$121\text{in}$

$\text{There is not enough information to solve the problem.}$

Explanation

An equilateral triangle has 3 equal sides. So, we will use the following formula:

where a is the length of one side of the triangle.

Now, we know the base of the triangle is 22in. Because all sides are equal, all sides are 22in. So, we can substitute into the formula:

$P = 3 \cdot 22\text{in}$

$P = 66\text{in}$

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

$69\text{in}$

$46\text{in}$

$58\text{in}$

$264\text{in}$

$128\text{in}$

Explanation

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

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

$P = 69\text{in}$

Find the perimeter of an equilateral triangle with a base of 22in.

$66\text{in}$

$242\text{in}$

$44\text{in}$

$121\text{in}$

$\text{There is not enough information to solve the problem.}$