# ACT Math : Equilateral Triangles

## Example Questions

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### Example Question #1 : Equilateral Triangles

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

Explanation:

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 .  Therefore, you can write the following equation:

This means that .

Now, the area of a triangle can be written:

, and based on our data, we can replace  with .  This gives you:

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

Solve for . Begin by multiplying each side by :

Divide each side by :

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

### Example Question #2 : Equilateral Triangles

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

Explanation:

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.

### Example Question #3 : Equilateral Triangles

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

Explanation:

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 , by , giving you .

### Example Question #4 : Equilateral Triangles

Find the perimeter of an equilateral triangle whose side length is

Explanation:

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

### Example Question #5 : Equilateral Triangles

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

Explanation:

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

### Example Question #6 : Equilateral Triangles

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

Explanation:

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

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

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

Explanation:

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 , , 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.

The side with length will be the height (opposite the 60 degree angle). The height is inches.

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

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

Explanation:

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:

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

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

Explanation:

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 .

As an equation, this is written:

Solving for , we get:

### Example Question #3 : How To Find The Height Of An Equilateral Triangle

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

is an equilateral triangle with side length  inches. What is the height of  ?