ACT Math : How to find the length of the side of an equilateral triangle

Study concepts, example questions & explanations for ACT Math

varsity tutors app store varsity tutors android store varsity tutors amazon store varsity tutors ibooks store

Example Questions

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

The height of an equilateral triangle is . Find the perimeter. 

Possible Answers:

Correct answer:

Explanation:

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

Equilateral

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

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, , 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  as our angle of interest, with the length  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.

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:

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:

Example Question #165 : Geometry

If an equilateral triangle has a height of  and an area of , what is the measure of one of its sides?

Possible Answers:

Correct answer:

Explanation:

For any triangle,  .

We know our height is  and our area is .

Therefore  so .

Example Question #166 : Geometry

An equilateral triangle has an area of  and a height of . What is the length of one of its sides?

Possible Answers:

Correct answer:

Explanation:

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:

In this case:

So the base is equal to 

Example Question #167 : Geometry

An equilateral triangle has a height of . What is the perimeter of this triangle?

Possible Answers:

Correct answer:

Explanation:

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    

 

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

In this case:

 

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

 

Learning Tools by Varsity Tutors

Incompatible Browser

Please upgrade or download one of the following browsers to use Instant Tutoring: