Pythagorean Theorem - GMAT Math Study Guide
The Pythagorean Theorem
A right triangle has one angle that is 90° (i.e., a right angle). The side opposite the right angle is called the hypotenuse, often labeled c, while the other two sides of the triangle (legs) are often labeled a and b.
For a right triangle with legs a and b and hypotenuse c:
Note: Which leg is a or b is irrelevant in the use of the Pythagorean Theorem.
Note: The Pythagorean Theorem only works for right triangles.
Putting the formula into words, the square of the length of one leg plus the square of the length of the other leg equals the square of the long-side opposite the right angle.
Examples Applying the Pythagorean Theorem
Given any two sides of a right triangle, it is possible to calculate what the length of the third side must be. For example:
What is the length of side c?
Pythagorean Theorem: a2
a = 3, b = 4, c = ?
9 + 16 = c2
25 = c2
c = 5
Note: When taking the square root, normally one must take the positive and negative value (i.e., ±5). However, since a negative length has no meaning when solving for the length of a side, the negative value is ignored.
Consider the following additional example:
If DGFE is a square with a side of length 5, what is the length of DF?
Since DGFE is a square, angle DEF is a right angle. Consequently, triangle DEF is a right triangle and the Pythagorean theorem can be used to find DF:
DE and EF = 5 since DGFE is a square
50 = (DF)2
DF = square root of 50
If a2 + b2 = c2, then the triangle is right
In other words, if a2 + b2 = c2, a triangle exists with sides a, b and c such that a right angle lies between the sides of length a and b
Consider the following example:
Is the triangle below a right triangle?
If the triangle is right, the length of its sides will fulfill the Pythagorean Theorem.
Plug in a = 5, b = 12, c = 13 and evaluate:
25 + 144 =? 169
169 = 169
The triangle is a right triangle.
There are three common relations between the sides of a right triangle. Recognizing these common side lengths can save considerable calculation time. Rather than using the Pythagorean theorem to calculate the missing side length, the length of the side can be determined by noticing the pattern.
3-4-5 triangles: 32 + 42 = 52
6-8-10, 9-12-15, and 12-16-20 triangles are simply multiples of the 3-4-5 rule.
5-12-13 triangles: 5 2 + 12 2 = 13 2
10-24-26 is another common way for this ratio to appear.
8-15-17 triangles:82 + 15 2 = 172
Consider the following example:
If a right triangle has legs 10 and 24, what is the length of the third side?
Instead of performing the calculation to find the third side, recognize that this triangle is a multiple of 5-12-13, where each side of the triangle is double the 5-12-13 pattern. Consequently, the third side is 2(13) = 26.
Special Right Triangles
Isosceles Right Triangle: 45°-45°-90°
As the above diagram indicates, a triangle with angles 45°-45°-90° will have sides in the ratio of 1-1-21/2
Bisected Equilateral Triangle: 30°-60°-90°
As the above diagram indicates, a triangle with angles 30°-60°-90° will have sides in the ratio of 1-31/2-2
Types of GMAT Problems
- Using Pythagorean Theorem in Complex Problems
Many GMAT questions will not simply give a right triangle with one missing side to calculate. Instead, a group of shapes will be given, with just enough information to eventually calculate a particular side, area, or angle. In these problems, it is best to look at what is being asked for. Next determine what is needed in order to calculate what is being asked for. Continue in this process until the given information is what is required. Then, from the given information, calculate through all of the various parts until the wanted quantity is found.
What is the area of the triangle BCD?
Correct Answer: E
- In order to calculate the area of BCD, the length of the base and height must be known.
- Any side can be chosen as the base, as long as the height is the line which goes through the angle opposite the base and is perpendicular to the base.
- Chose CB to be the base, as it can be calculated by use of the Pythagorean Theorem.
Triangle ABC, which is a right triangle, has sides 6-?-10. This is a multiple of the 3-4-5 rule. Thus, BC is 8 (=4*2).
- The height of triangle CDB will be the length of the line from D to BC. This height line is by definition perpendicular to BC. The height line will be the same length as BE since a square is formed with points D, E, B, and the intersection of a perpendicular line from D to line CB.
- Calculate the length of BE using the Pythagorean Theorem.
5-?-13 is another common right triangle with lengths 5-12-13
Thus, BE is 12
52 + x2 = 132
25 + x2 = 169
x2 = 144
x = 12
- With the base BC = 8 and the height BE = 12, calculate the area:
A = .5 * base * height
A = .5 * 8 * 12 = 48