A Guide to the GRE/Pythagorean Theorem

The Pythagorean Theorem
In a right triangle with sides a, b and c, a2 + b2 = c2, c being the longest side.

In this triangle, a2 + b2 = c2 and thus:

a =

b =

c =

There are a limited number of “Pythagorean Triples” or groups of integers which fit the formula.

The GRE tends to have questions which work out to integers; thus, keep an eye out for the “3-4-5” triangle and the “5-12-13” triangle.

The 3-4-5 Triangle

The squares of 3 and 4 add up to the square of 5, 		thus, the “3-4-5” triangle is common on the GRE. It may also be in the form of a “6-8-10” triangle or 		a “9-12-15” triangle, or another triangle similarly 				increased.

The 5-12-13 Triangle

In a similar manner the 3-4-5 triangle, this triangle has a series of integers which comport with the Pythagorean Theorem. This triple may also be increased to 10-24-26 and so on.

There are two special right triangles, the proportions of which can be inferred by their angle measurements.

A 45º-45º-90º triangle has sides in the ratio of 1 - 1 -

Thus if the shorter sides are 10, the longer side is

A 30º-60º-90º triangle has sides in the ratio of 1 -- 2.

Thus if the shortest side has a length of 5, the two other sides have lengths ofand 10.

Indeed, the proportions of every right triangle can be determined by the measurement of one of its angles based on this theorem. Mathematicians and engineers have developed tables of the proportions for these angles and have designated functions for them, such as sin and cos. This field of mathematics is known as “trigonometry” and is not tested on the GRE.

Practice
1.

What is the value of q?

2.

Solve for r.

3.

Determine the value of s.

Answers to Practice Questions
1.

In a right triangle with sides a, b and c, a2 + b2 = c2, where c is the longest side. Thus, q is equal to or

2.

In a right triangle with two equal sides, the longer side equals the shorter side multiplied by Thus, the shorter side equals the longer side multiplied by This means r equals 8 times this proportion, which equals or

3. 24

In a right triangle with sides a, b and c, a2 + b2 = c2, where c is the longest side. Thus, s equals which equals or 24. This is an example of the “Pythagorean Triple” of 5-12-13, in this case multiplied by 2.