7.7 - Justifying the Laws
- Author:
- Ms. Dougherty-Samowitz, John Golden
The Pythagorean theorem makes a claim about the relationship between the areas of the three squares drawn on the sides of a right triangle: the sum of the area of the squares on the two legs is equal to the area of the square on the hypotenuse. We generally state this relationship algebraically as , where it is understood that a and b represent the length of the two legs of the right triangle, and c represents the length of the hypotenuse.
What about non-right triangles? Is there a relationship between the areas of the squares drawn on the sides of a non-right triangle? (Note: The following proof is based on The Illustrated Law of Cosines, by Don McConnell)
The diagram below shows an acute triangle with squares drawn on each of the three sides. The three altitudes of the triangle have been drawn and extended through the squares on the sides of the triangle. The altitudes divide each square into two smaller rectangles.