The Pythagorean Theorem is a trigonometric identity that relates the lengths of the three sides of a 90 degree triangle. A 90° triangle is one in which one of its three interior angles is equal to 90° with the sum of all angles being 180°. The Pythagorean Theorem states simply that the square of the hypotenuse (the longest leg of the triangle, we call this c) equivalent to the summation of the squares of each of the two smaller legs (we call these a and b). In other terms, c^{2} = a^{2 }+ b^{2}. With this relationship, if any two values are known the other can be determined. It should be noted that the each angle in the triangle is proportional to its adjacent length.

Two common yet special 90 degree triangle cases exist. The first is a 45-45 triangle. This is a triangle in which two of its sides (both being less than the length of the hypotenuse) are equal and each have an adjacent angle of 45°. The second is a 60-30 triangle. In this triangle, the smallest leg has an adjacent angle of 30° and is half the length of the hypotenuse which has an adjacent angle of 90° with the second largest leg and it having an adjacent angle of 60°.

Let’s see an example to understand this concept.

Say we have a right triangle in which we know the hypotenuse is 60 units and one of the legs is 52 units. If we apply the Pythagorean Theorem, we obtain the following:

c^{2} = a^{2 }+ b^{2 }(Pythagorean Theorem)

(60)^{2} = (52)^{2 }+ b^{2 }(Substitute our known values)

b^{2} = (60^{2 }- 52^{2})^{1/2} (Rearrange the equation to solve in terms of the unknown value)

b^{2} = (896)^{ 1/2}

b = 30

So the length of our other leg is 30 units and we can see that this unit is half the length of the hypotenuse and by our definition above it is a 60-30 triangle. We can use Trigonometry to verify these angles are correct.

Angle A = tan^{-1}(a/b) = tan^{-1}(52/30) = 60° (this is the angle that is adjacent to leg A)

We know that the sum of all interior angles is 180° and if we subtract the 90° and 60° angles we are left with 30°, this is the angle measure of Angle B which is adjacent to leg B.

RESOURCE:

Bittinger, M.L., Beecher, J.A., Ellenbogen, D., Penna, J.A. (2001). Precalculus: Graphs & Models (2^{nd} Ed.). United States of America: Addison Wesley Longman, Inc.