A Generalization of the Pythagorean Theorem

In triangle ABC, if C is a right angle, then the Pythagorean Theorem tells us c² = a² + b²:

But what if C is not a right angle?

First suppose C is less than 90^o. Draw an altitude from B to AC:

By the Pythagorean Theorem, c² = x² + h², and h² = a² – y², so c² = x² + a² – y².  But x + y = b, so x = b – y, and therefore c² = (b – y)² + a² – y² = a² + b² – 2by.  Now

So finally:

Now suppose C is greater than 90^o.  By similar reasoning, we again find that

This makes use of the fact that cos(180^o – θ) = –cos θ as can be seen when θ is in standard position:

 

The following examples show how the Law of Cosines can be used to find missing parts of triangles.

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