# GRE Math : Plane Geometry

## Example Questions

### Example Question #21 : Triangles

What is a possible value for the length of the missing side?

Explanation:

For a triangle where the length of two sides,  and , is the only information known, the third side, , is limited in the following matter:

For the triangle given:

.

Both choices A and B satisfy this criteria.

### Example Question #2 : How To Find The Length Of The Side Of An Acute / Obtuse Triangle

A triangle has sides of lengths  and

Quantity A: The length of the missing side.

Quantity B:

Quantity A is greater.

The relationship cannot be determined.

Quantity B is greater.

The two quantities are equal.

Quantity A is greater.

Explanation:

If two sides of a triangle are known and all angles are unknown, the length of the third side is limited by the difference and sum of the other two sides.

The missing side must be greater than .

Quantity A is greater.

### Example Question #3 : How To Find The Length Of The Side Of An Acute / Obtuse Triangle

The lengths of two sides of a triangle are  and .

Quantity A: The length of the missing side.

Quantity B:

Quantity B is greater.

Quantity A is greater.

The relationship cannot be determined.

The two quantities are equal.

The relationship cannot be determined.

Explanation:

Seeing the sides  and  may bring to mind a  triangle. However, we've been told nothing about the angles of the triangle. It could be right, or it could be obtuse or acute.

Since the angles are unknown, the side is bounded as follows:

There are plenty of potential lengths that fall above and below . The relationship cannot be determined.

### Example Question #4 : How To Find The Length Of The Side Of An Acute / Obtuse Triangle

A triangle has sides  and

Quantity A: The length of the missing side.

Quantity B:

Quantity A is greater.

The relationship cannot be determined.

The two quantities are equal.

Quantity B is greater.

Quantity B is greater.

Explanation:

If two sides of a triangle are known and the angles of the triangle are unknown, the length of the missing side is limited by the difference and sum of the other two sides.

For a triangle with sides  and , there is no way a side could be .

Quantity B is greater.

### Example Question #1 : Equilateral Triangles

Find the perimeter of an equilateral triangle with a height of .

None of the answer choices are correct.

Explanation:

Perimeter is found by adding up all sides of the triangle. All sides in an equilateral triangle are equal, so we need to find the value of just one side to know the values of all sides.

The height of an equilateral triangle divides it into two equal 30:60:90 triangles, which will have side ratios of 1:2:√3. The height here is the √3 ratio, which in this case is equivalent to 8, so to get the length of the other two sides, we put 8 over √3 (8/√3) and 2 * 8/√3 = 16/√3, which is the hypotenuse of our 30:60:90 triangle.

The perimeter is then 3 * 16/√3, or 48/√3.

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

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

Explanation:

By having a height in an equilateral triangle, the angle is bisected therefore creating two  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

Multiply top and bottom by  to get rid of the radical.

Since each side is the same and there are three sides, we just multply the answer by three to get

### Example Question #32 : Triangles

If area of equilateral triangle is , what is the perimeter?

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 . Since we have three equal sides, we just multply  by three to get  as the final answer.

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

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

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  as the final answer.

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

If the height of the equilateral triangle is , then what is the length of a side of an equilateral triangle?

Explanation:

By having a height in an equilateral triangle, the angle is bisected therefore creating two  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

Multiply top and bottom by  to get rid of the radical.

### Example Question #2 : Equilateral Triangles

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