Equilateral Triangles - ACT Math

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Question

A circle contains 6 copies of a triangle; each joined to the others at the center of the circle, as well as joined to another triangle on the circle’s circumference.

The circumference of the circle is $4\pi$

What is the area of one of the triangles?

Answer

The radius of the circle is 2, from the equation circumference $=2r\pi$ . Each triangle is the same, and is equilateral, with side length of 2. The area of a triangle $=\frac{1}{2}bh$

To find the height of this triangle, we must divide it down the centerline, which will make two identical 30-60-90 triangles, each with a base of 1 and a hypotenuse of 2. Since these triangles are both right traingles (they have a 90 degree angle in them), we can use the Pythagorean Theorem to solve their height, which will be identical to the height of the equilateral triangle.

$a^{2}+b^{2}=c^{2}$

We know that the hypotenuse is 2 so $2^{2}=4$ . That's our $c^{2}$ solution. We know that the base is 1, and if you square 1, you get 1.

Now our formula looks like this: $a^{2}+1=4$ , so we're getting close to finding .

Let's subtract 1 from each side of that equation, in order to make things a bit simpler: $a^{2}+0=3 \rightarrow a^{2}=3$

Now let's apply the square root to each side of the equation, in order to change $a^{2}$ into : $a=\sqrt{3}$

Therefore, the height of our equilateral triangle is $\sqrt{3}$

To find the area of our equilateral triangle, we simply have to multiply half the base by the height: $\frac{2}{2}\sqrt{3}=1*\sqrt{3}=\sqrt{3}$

The area of our triangle is $\sqrt{3}$

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Question

What is the area of an equilateral triangle with a side length of 5?

Answer

Note that an equilateral triangle has equal sides and equal angles. The question gives us the length of the base, 5, but doesn't tell us the height.

If we split the triangle into two equal triangles, each has a base of 5/2 and a hypotenuse of 5.

Therefore we can use the Pythagorean Theorem to solve for the height:

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

$\frac{25}{4}+b^2=25$

$b^2=\frac{75}{4}$

$b=\sqrt{\frac{3\times 25}{2\times 2}}=\frac{5\sqrt{3}}{2}$

Now we can find the area of the triangle:

$Area=\frac{1}{2}\times base\times height=\frac{1}{2}\times 5\times \frac{5\sqrt{3}}{2}=\frac{25\sqrt{3}}{4}$

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Question

What is the area of an equilateral triangle with sides of length ?

Answer

While you can very quickly compute the area of an equilateral triangle by using a shortcut formula, it is best to understand how to analyze a triangle like this for other problems. Let's consider this. The shortcut will be given below.

Recall that from any vertex of an equilateral triangle, you can drop a height that is a bisector of that vertex as well as a bisector of the correlative side. This gives you the following figure:

Notice that the small triangles within the larger triangle are both triangles. Therefore, you can create a ratio to help you find .

The ratio of to is the same as the ratio of to $\sqrt{3}$ .

As an equation, this is written:

$\frac{10}{h}=\frac{1}{\sqrt{3}}$

Solving for , we get: $h=10\sqrt{3}$

Now, the area of the triangle is merely $\frac{1}{2}bh$ . For our data, this is: $\frac{1}{2}2010\sqrt{3}$ or $100\sqrt{3}$ .

Notice that this is the same as $\frac{b^2\sqrt{3}}{4}$ . This is a shortcut formula for the area of equilateral triangles.

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Question

What is the area of an equilateral triangle with a perimeter of ?

Answer

Since an equilateral triangle is comprised of sides having equal length, we know that each side of this triangle must be $\frac{51}{3}$ or . While you can very quickly compute the area of an equilateral triangle by using a shortcut formula, it is best to understand how to analyze a triangle like this for other problems. Let's consider this. The shortcut will be given below.

Recall that from any vertex of an equilateral triangle, you can drop a height that is a bisector of that vertex as well as a bisector of the correlative side. This gives you the following figure:

Notice that the small triangles within the larger triangle are both triangles. Therefore, you can create a ratio to help you find .

The ratio of to is the same as the ratio of to $\sqrt{3}$ .

As an equation, this is written:

$\frac{8.5}{h}=\frac{1}{\sqrt{3}}$

Solving for , we get: $h=8.5\sqrt{3}$

Now, the area of the triangle is merely $\frac{1}{2}bh$ . For our data, this is: $\frac{1}{2}178.5\sqrt{3}$ or $72.25\sqrt{3}$ .

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Question

What is the area of a triangle with a circle inscribed inside of it, in terms of the circle's radius R?

Answer

Draw out 3 radii and 3 lines to the corners of each triangle, creating 6 30-60-90 triangles.

See that these 30-60-90 triangles can be used to find side length.

$S = R2\sqrt{3}$

Formula for side of equilateral triangle is

$\frac{\sqrt{3}}{4} * S^2$ .

Now substitute the new equation that is in terms of R in for S.

$=\frac{\sqrt{3}}{4}(R2\sqrt{3})^2$

$=\frac{\sqrt{3}}{4} 12 * R^{2}$

$= 3\sqrt{3}R^2$

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Question

What is the area of a triangle inscribed in a circle, in terms of the radius R of the circle?

Answer

Draw 3 radii, and then 3 new lines that bisect the radii. You get six 30-60-90 triangles.

These triangles can be used ot find a side length $S = R\sqrt{3}$ .

Using the formula for the area of an equilateral triangle in terms of its side, we get

$\frac{\sqrt{3}}{4}S^2$

$\frac{\sqrt{3}}{4}(R\sqrt3)^2 =$

$3\frac{\sqrt{3}}{4} * R^2$

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Question

What is the height of an equilateral triangle with a side length of 8 in?

Answer

An equilateral triangle has three congruent sides, and is also an equiangular triangle with three congruent angles that each meansure 60 degrees.

To find the height we divide the triangle into two special 30 - 60 - 90 right triangles by drawing a line from one corner to the center of the opposite side. This segment will be the height, and will be opposite from one of the 60 degree angles and adjacent to a 30 degree angle. The special right triangle gives side ratios of , $x\sqrt{3}$ , and . The hypoteneuse, the side opposite the 90 degree angle, is the full length of one side of the triangle and is equal to . Using this information, we can find the lengths of each side fo the special triangle.

$8=2x\rightarrow 4=x\rightarrow 4\sqrt{3}=x\sqrt{3}$

The side with length $x\sqrt{3}$ will be the height (opposite the 60 degree angle). The height is $4\sqrt{3}$ inches.

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Question

Find the height of a triangle if all sides have a length of .

Answer

Draw a vertical line from the vertex. This will divide the equilateral triangle into two congruent right triangles. For the hypothenuse of one right triangle, the length will be . The base will have a dimension of . Use the Pythagorean Theorem to solve for the height, substituting in for , the length of the hypotenuse, and for either or , the length of the legs of the triangle:

$b=\sqrt3$

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Question

What is the height of an equilateral triangle with sides of length ?

Answer

Recall that from any vertex of an equilateral triangle, you can drop a height that is a bisector of that vertex as well as a bisector of the correlative side. This gives you the following figure:

Notice that the small triangles within the larger triangle are both triangles. Therefore, you can create a ratio to help you find .

The ratio of to is the same as the ratio of to $\sqrt{3}$ .

As an equation, this is written:

$\frac{6}{h}=\frac{1}{\sqrt{3}}$

Solving for , we get: $h=6\sqrt{3}$

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Question

The Triangle Perpendicular Bisector Theorem states that the perpendicular bisector of an equilateral triangle is also the triangle's height.

$\Delta ABC$ is an equilateral triangle with side length inches. What is the height of $\Delta ABC$ ?

Answer

To calculate the height of an equilateral triangle, first draw a perpendicular bisector for the triangle. By definition, this splits the opposite side from the vertex of the bisector in two, resulting in two line segments of length inches. Since it is perpendicular, we also know the angle of intersection is $90^{\circ}$ .

So, we have a new right triangle with two side lenghts and for the hypotenuse and short leg, respectively. The Pythagorean theorem takes over from here:

---> $x^2 = 48 = 4\sqrt{3}$

So, the height of our triangle is $4\sqrt{3}$ .

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Question

The height of an equilateral triangle is $6\sqrt3:cm$ . Find the perimeter.

Answer

From the given information, we can draw the following picture:

Since this is an equilateral triangle, all three of its angles will be congruent. All three of its sides will also be congruent.

The dotted line, representing the height, bisects not only the base of the triangle but also the apical angle. This means that the two smaller right triangles created by the dotted line have bases that are half the side length of the equilateral triangle with apical angle measures of $30^{\circ}$ .

This creates two smaller 30/60/90 special right triangles. This problem can be solved in one of two ways:

1. You can use the derived side ratio for 30/60/90 triangles, $h:2h:\sqrt3\cdot h$ , to solve for the length of one of the equilateral triangle's sides.

OR

2. You can use trig functions to solve for the length of one of the equilateral triangle's sides.

Using trig functions, we can use $60^{\circ}$ as our angle of interest, with the length $6\sqrt3$ and the hypotenuse being our mystery side.

Going back to SOH CAH TOA, we can determine that we will be using sine because we have the information available for the side opposite ("O") of our angle of interest and are looking for the hypotenuse ("H"). This means we will be using the "SOH" part of SOH CAH TOA—sine.

$sin(x)=\frac{opposite}{hypotenuse}$

Substituting in the problem's values, we can solve for the length of the hypotenuse of one of the two smaller triangles, which is the length of one side of the equilateral triangle:

$sin(60^{\circ})=\frac{6\sqrt{3}}{h}$

$h \cdot sin(60^{\circ})=6\sqrt{3}$

$h= \frac{6\sqrt{3}}{sin(60^\circ)}$

Remember that the problem asks for the triangle's perimeter. That means that we need to multiply the length of one of its edges by three to find the final answer:

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Question

If an equilateral triangle has a height of $\sqrt{12}$ and an area of $2\sqrt{12}$ , what is the measure of one of its sides?

Answer

For any triangle, $area=\frac{(base\cdot height)}{2}$ .

We know our height is $\sqrt{12}$ and our area is $2\sqrt{12}$ .

Therefore $2\sqrt{12}=\frac{(base\cdot \sqrt{12})}{2}$ so $base=\frac{4\sqrt{12}}{\sqrt{12}}=\frac{4\cdot2\sqrt{3}}{2\sqrt{3}}=4$ .

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Question

An equilateral triangle has an area of $\frac{3\sqrt{3}}{2}$ and a height of . What is the length of one of its sides?

Answer

Since this triangle is equilateral, all of the sides (including the base) are equal.

Use the formula for area of a triangle to solve for the base:

$\frac{base \times height }{2}$

In this case:

$\frac{base \times 3}{2}=\frac{3\sqrt{3}}{2}$

So the base is equal to $\sqrt{3}$

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Question

An equilateral triangle has a height of $3\sqrt{3}$ . What is the perimeter of this triangle?

Answer

Since this is an equilateral triangle, all of the sides and all of the angles would be equal (60 degrees each since the 3 angles must add up to 180 degrees).

1. The height splits the triangle's base in half and creates two right triangles. Create expressions for the length of the two unknown sides in this new triangle:

The value we are looking for is the side length so we'll make that

The base of the new triangle is half the length of a side so we'll make that $\frac{x}{2}$

2. Use the Pythagorean Theorem to find the value of or the side length:

$a^{2}+b^{2}=c^{2}$

In this case:

$\left(\frac{x}{2}\right)^{2}+(3\sqrt{3})^{2}=x^{2}$

$\left(\frac{x}{2}\right)^{2}+(3)^{2}(\sqrt{3})^{2}=x^{2}$

$\frac{x^{2}}{4}+27=x^{2}$

$(4)\frac{x^{2}}{4}+(4)(27)=4x^{2}$

$x^{2}+108=4x^{2}$

$3x^{2}=108$

$x^{2}=36$

3. Multiply the side length you found by 3 to get the perimeter:

$6\times3=18$

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Question

Find the perimeter of an equilateral triangle given side length of 2.

Answer

To solve, simply multiply the side length by 3 since they are all equal. Thus,

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Question

What is the perimeter of an equilateral triangle with an area of $56.25\sqrt{3}$ ?

Answer

Recall that from any vertex of an equilateral triangle, you can drop a height that is a bisector of that vertex as well as a bisector of the correlative side. This gives you the following figure:

Notice that the small triangles within the larger triangle are both triangles. Therefore, you can create a ratio to help you find .

The ratio of the small base to the height is the same as $1:\sqrt{3}$ . Therefore, you can write the following equation:

$\frac{0.5b}{h}=\frac{1}{\sqrt{3}}$

This means that $0.5b\sqrt{3}=h$ .

Now, the area of a triangle can be written:

$A=\frac{1}{2}bh$ , and based on our data, we can replace with $\frac{b\sqrt{3}}{2}$ . This gives you:

$56.25\sqrt{3}=\frac{1}{2}b\frac{b\sqrt{3}}{2}=\frac{b^2\sqrt{3}}{4}$

Now, let's write that a bit more simply:

$56.25\sqrt{3}=\frac{b^2\sqrt{3}}{4}$

Solve for . Begin by multiplying each side by :

$225\sqrt{3}=b^2\sqrt{3}$

Divide each side by $\sqrt3$ :

Finally, take the square root of both sides. This gives you . Therefore, the perimeter is .

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Question

An equilateral triangle with a perimeter of has sides with what length?

Answer

An equilateral triangle has 3 equal length sides.

Therefore the perimeter equation is as follows,

.

So divide the perimeter by 3 to find the length of each side.

Thus the answer is:

$\frac{48}{3}=s$

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Question

Jill has an equilateral triangular garden with a base of and one leg with a length of , what is the perimeter?

Answer

Since the triangle is equilateral, the base and the legs are equal, so the first step is to set the two equations equal to each other. Start with , add to both sides giving you . Subtract from both sides, leaving . Finally divide both sides by , so you're left with . Plug back in for into either of the equations so that you get a side length of . To find the perimeter, multiply the side length $\left ( 34\right )$ , by , giving you .

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Question

Find the perimeter of an equilateral triangle whose side length is

Answer

To find perimeter of an quilateral triangle, simply multiply the side length by . Thus,

$P=s\cdot3=2.1\cdot3=6.3$

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Question

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

Answer

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

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