GRE Exam: A Simple Guide To Solving Geometry Problems - Triangles

Triangles are the GRE’s favorite shape. They appear not only in questions that explicitly ask you about triangles, but also in disguised form on questions addressing other polygons, such as squares or rectangles. As you may recall from high school, there are numerous properties associated with triangles.

Let’s look below at the properties you need to master for the GRE.

Basic Properties of Triangles

Basic Property 1: The sum of the internal angles in a triangle equals 180. Thus in the preceding diagram, a + b + c = 180.

Example: In the figure above, x = y = 2z. What is x?
SOLUTION: Since x, y, and z are the interior angles of a triangle, x + y + z = 180. Use the given information to express all variables in terms of z:

Since x = 2z, x = 2(36) = 72.

Basic Property 2: The length of any given side of a triangle must be greater than the difference of the other two side lengths and less than the sum of the other two side lengths.

For this question, indicate all the answer choices that apply.
Example: If a triangle has side lengths of 4 and 7, which of the following could be the length of the third side of the triangle?
3
4
5
9
11
13

SOLUTION: The third side of the triangle must be greater than (7 – 4) and less than (7 + 4). Thus the length of the third side must be between 3 and 11. Of the choices, the length could be 4, 5, or 9.

Basic Property 3: The greater the measurement of a triangle’s angle, the greater the length of the corresponding side.
An angle’s corresponding side is the side opposite that angle. In the following triangle, each angle’s corresponding side is indicated by the arrows.

Note that this relationship also works in reverse: the greater the length of a side, the greater the measurement of the corresponding angle. It is also important to note that this information only provides a relationship between the corresponding sides and angles of a triangle.

Without additional information, you cannot infer how much greater one side or angle is than another side or angle.


Isosceles Triangles
An isosceles triangle is any triangle that has two equal angles. Since the two angles are equal, the corresponding sides will also be equal.

Thus as you saw with the Basic Property 3, the relationship works in reverse. If a triangle has two equal sides, it will be isosceles, and the angles opposite those sides will be equal.

This information is helpful because when you are working with an isosceles triangle, you can assign the same variable to different angles or sides.

 

Example: In the figure above, side YZ = side XY. If x = 2y, then y = ?
SOLUTION: Since YZ and XY are equal, their corresponding angles must be equal.

Thus x = z.

The sum of the interior angles of a triangle is 180, so x + x + y = 180.

Substitute 2y for x:

However, from the fact that a triangle is isosceles, you cannot necessarily infer which sides or angles are equal.


SOLUTION: Since triangle ABC is isosceles, it is possible that angle C = 40, in which case the two quantities are equal. However, it can also be the case that angle B = 40 and angle C = 100, in which case angle C > 40.

Thus, the relationship cannot be determined. The answer is D.

Equilateral Triangles
An equilateral triangle is a triangle in which all angles are equal and all sides are equal.

Since the angle measurements are the same and their sum is 180, each angle in an equilateral triangle measures 60 degrees.

 

Perimeter of a Triangle
The perimeter of a triangle is the sum of all the side lengths.

The perimeter of the preceding triangle is 3 + 5 + 7 = 15.

Area of a Triangle
The area of a triangle refers to the amount of space within the triangle.

Area is expressed in square units, such as cm2 (square centimeters), in.2 (square inches), and so on.

The formula for the area of a triangle is .

Note that the base (b) and height (h) must be perpendicular to each other.

Look at the following figure:

In this example, the base is 8 and the height is 7.

You know that the height is 7 because the line segment drawn from angle C creates a perpendicular angle when it intersects the base, AB.

The area of this triangle will thus be .

In the example, AB was the base of the triangle. However, any side can be the base of a triangle.

The height will be defined as the perpendicular line from the angle opposite the base.

Next, you will see the same triangle as earlier, but now BC is the base.

Now, to calculate the area:

.

Sometimes you will have to extend the base to determine the corresponding height:

In this example, the base is 6.

To draw the height, it was necessary to extend base BC until it intersected the height.

The area of this triangle is .

Right Triangles
The GRE’s favorite shape is a right triangle. A right triangle is any triangle that has a 90-degree angle. The 90-degree angle is formed at the intersection of the two shorter sides, which are called the legs.

The side opposite the 90-degree angle is the longest side of a right triangle. It is called the hypotenuse.

The formula for the area of any triangle applies to right triangles, but in right triangles, determining the area is easier than with other triangles. Why? Because the base and height will simply be the two sides that form the 90-degree angle.

This is so because these two sides are perpendicular, meaning that one leg can be considered the base and the other leg can be considered the height (it doesn’t matter which leg you call the base and which leg you call the height).

The area of the preceding triangle is 48. If the length of AB = 8, then what is the length of BC?

SOLUTION: The area of a right triangle is .

Substitute 8 for (leg 1) and BC for leg 2:


The Pythagorean Theorem
Recall from Basic Property 3 that the greater the angle of a triangle, the longer the corresponding side. Since the 90 degree angle in a triangle must be the largest angle of the triangle, its corresponding side must be the longest side of the triangle. This side is called the hypotenuse. The two sides forming the 90 degree angle are called the legs.

The Pythagorean theorem provides a relationship that always holds true between the sides of a right triangle. If a = the length of one leg, b = the length of the other leg, and c = the length of the hypotenuse, then:

What is the perimeter of the triangle above?
13
17
25
30
42

SOLUTION: To determine the perimeter, solve for side BC, which is the hypotenuse.

You can use the Pythagorean theorem to solve for the hypotenuse:

Thus the perimeter is 5 + 12 + 13 = 30.

Pythagorean Triplets
There are certain combinations of right-triangle side lengths that occur throughout the GRE.

Though you can always use the Pythagorean theorem in these situations, you will save precious time by memorizing these triplets and their multiples:


 

What is the perimeter of the figure above?
SOLUTION: To determine the perimeter, you need to add up all the sides. You know all the side-lengths except CD.

To solve for CD, first solve for AC. The legs of right triangle ABC have lengths of 3 and 4, so AC = 5. If AC = 5 and AD = 12, then CD = 13.

The perimeter of the figure is thus 3 + 4 + 12 + 13 = 32.

However, be careful about assuming that any right triangle with two side lengths from the Pythagorean triplets will necessarily conform to the preceding list.

The triplets only apply in situations where the largest value of the triplet is the length of the hypotenuse:

SOLUTION: Even though ABC is a right triangle with two sides of 3 and 4, it is not a Pythagorean triplet. The side with length 4 is the hypotenuse, which means that the length of AC must be less than 4. Thus Quantity B is greater.

Isosceles Right Triangles and the Diagonal of a Square
In addition to the Pythagorean triplets, there are two other triangle combinations that you will need to know for the GRE: isosceles right triangles and 30-60-90 triangles.
An isosceles right triangle is any right triangle in which the lengths of the legs are equal.

Since the lengths of the legs are equal, their corresponding angles will also be equal, with each having a measurement of 45 degrees.

Thus another term for an isosceles right triangle is a 45-45-90 triangle.

The legs of every isosceles right triangle will have a specific ratio that you should memorize:

It is important to note that the preceding combination only specifies a ratio, and not actual values.

For example, if you are told that the leg length of an isosceles triangle is 5, then the hypotenuse is .

Or if the leg length is 7, then the hypotenuse is .

The best way to think about the relationships of the leg lengths is that the hypotenuse will be the leg.

One commonly tested fact about 45-45-90 triangles is that the diagonal of a square will form two 45-45-90 triangles.

This is helpful because you can use the diagonal of the square to solve for the side lengths of the square and vice versa.

 

What is the area of a square with a diagonal of length 20?


SOLUTION: To solve for the area, you need the length of a side.

The length of the side will be the leg of an isosceles right triangle with a hypotenuse of 20.

Let x = leg length:

30-60-90 Triangles and the Equilateral Triangle
The other type of special right triangle you need to master is the 30-60-90 triangle.

To understand the properties of a 30-60-90 triangle, look at what happens when you draw the height of an equilateral triangle:

Fact 1: The height of an equilateral triangle will cut the base in half.
Fact 2: The resulting smaller triangles will have degree measurements of 30-60-90.
Fact 3: The sides of the 30-60-90 triangle will be in the following ratio, which you must memorize:

Finally, as was the case with 45-45-90 triangles, with 30-60-90 triangles, the side relationships only specify ratios, not values.

What is the perimeter of triangle BCD in the figure above?

SOLUTION: To determine the perimeter, you must determine the side lengths of BCD. Note that side BC is the hypotenuse of the 45-45-90 triangle ABC. Thus .

Since BC is the shorter leg of 30-60-90 triangle BCD, the longer leg, CD, will equal , and the hypotenuse, BD, will equal 10 × 2 = 20.

The perimeter of BCD is thus .