GRE Math : How to find the height of an equilateral triangle

Study concepts, example questions & explanations for GRE Math

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Example Questions

Example Question #9 : Equilateral Triangles

One side of an equilateral triangle is equal to 

Quantity A: The area of the triangle.

Quantity B: 

Possible Answers:

Quantity A is greater.

The relationship cannot be determined.

Quantity B is greater.

The two quantities are equal.

Correct answer:

Quantity B is greater.

Explanation:

To find the area of an equilateral triangle, notice that it can be divided into two  triangles:

Equilateral triangle

The ratio of sides in a  triangle is , and since the triangle is bisected such that the  degree side is , the  degree side, the height of the triangle, must have a length of .

The formula for the area of the triangle is given as:

So the area of an equilateral triangle can be written in term of the lengths of its sides as:

For this particular triangle, since , its area is equal to .

If the relation between ratios is hard to visualize, realize that 

 

Example Question #10 : Equilateral Triangles

If the area of an equilateral triangle is , what is the height of the triangle?

Possible Answers:

Correct answer:

Explanation:

The area of an equilateral triangle is .

So let's set-up an equation to solve for 

 Cross multiply.

 

The  cancels out and we get .

Then take square root on both sides and we get . To find height, we need to realize by drawing a height we create   triangles. 

The height is opposite the angle . We can set-up a proportion. Side opposite  is  and the side of equilateral triangle which is opposite  is .

 Cross multiply.

 Divide both sides by 

 

We can simplify this by factoring out a  to get a final answer of 

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

Quantity A: The height of an equilateral triangle with an area of 

Quantity B: 

Which of the following is true?

 

Possible Answers:

Quantity B is greater.

The relationship cannot be determined.

The two quantities are equal.

Quantity A is greater.

Correct answer:

Quantity B is greater.

Explanation:

This problem requires a bit of creative thinking (unless you have memorized the fact that an equilateral triangle always has an area equal to its side length times .

Consider the equilateral triangle:

Equilateral8

Since this kind of triangle is a species of isoceles triangle, we know that we can drop down a height from the top vertex. This will create two equivalent triangles, one of which will look like:

Equilateral8 2

This gives us a 30-60-90 triangle. We know that for such a triangle, the ratio of the side across from the 30-degree angle to the side across from the 60-degree angle is:

We can also say, given our figure, that the following equivalence must hold:

Solving for , we get:

Now, since , we know that  must be smaller than . This means that  or . Quantity B is larger than quantity A.

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

Quantity A: The height of an equilateral triangle with perimeter of .

Quantity B: 

Which of the following is true?

Possible Answers:

Quantity A is larger.

The two quantities are equal.

Quantity B is larger.

The relationship cannot be determined.

Correct answer:

Quantity B is larger.

Explanation:

If the perimeter of our equilateral triangle is , each of its sides must be  or . This gives us the following figure:

Equilateral9

Since this kind of triangle is a species of isoceles triangle, we know that we can drop down a height from the top vertex. This will create two equivalent triangles, one of which will look like:

Equilateral9 2

 

This gives us a 30-60-90 triangle. We know that for such a triangle, the ratio of the side across from the 30-degree angle to the side across from the 60-degree angle is:

Therefore, we can also say, given our figure, that the following equivalence must hold:

Solving for , we get:

Now, since , we know that  must be smaller than . This means that  or 

Therefore, quantity B is larger than quantity A.

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