Proof of Pythagoras' Theorem
Introduction
The proof is further down this page here. First an introduction. Remember the 3, 4, 5 triangle used to introduce Pythagoras' Theorem? Suppose we take four of these triangles and join them together to make this figure:
The total area of this figure can be calculated in two ways. One way would be to add together the area of the small square in the middle with the areas of the four triangles. Another way is to calculate the area of the large square which has 7cm sides. Whichever way we choose, we'll get the same area.
(area of small square) + (area of four triangles)
= (length × width) + (4 × ½ × base × height)
= (5 × 5) + (4 × ½ × 4 × 3)
= 25 + 24
= 49 cm2
If we calculate the area of the large square we get 7 × 7, which is also 49 cm2
If we carry out the same exercise as above using a, b and c instead of 3, 4 and 5, we will finish by proving not that 49 = 49 (which we all know!), but that
a2 + b2 = c2. Consider this figure formed by joining four identical (or, to use the proper term, congruent) right angled triangles:
(area of small square) + (area of four triangles)
= (length × width) + (4 × ½ × base × height)
= (c2) + (4 × ½ × a × b)
= c2 + 2ab
(area of large square)
= (a + b)2
= a2 + 2ab + b2
By looking at the figure, we know that:
(area of large square) = (area of small square) + (area of four triangles) so:
a2 + 2ab + b2 = c2 + 2ab .
By just subtracting 2ab from both sides we get what we want, Pythagoras' Theorem:
a2 + b2 = c2