Pythagoras Theorem: Deep Dive
\color{blue}{9} + \color{green}{16} = \color{orange}{25}Transcript
Three squares, one right angle, and a relationship that has been proved in hundreds of different ways. Let's find out why it holds. Here is a right triangle with legs three and four. The side opposite the right angle is called the hypotenuse. Leg a is vertical, and it is the blue side.
Leg b is horizontal, and it is the green side. The hypotenuse c, in orange, is always the longest side, because it faces the largest angle. The theorem is really about area. Build a square on leg a and count its unit squares.
Now a square on leg b. Four times four gives sixteen unit squares. Finally a square on the hypotenuse. Its area should equal the other two squares added together. Nine plus sixteen is twenty-five, and the big square has side five. Area is conserved, so the squares on the legs sum to the square on the hypotenuse.
\color{blue}{9} + \color{green}{16} = \color{orange}{25}In symbols, with side lengths a, b and c, the Pythagorean equation says a squared plus b squared equals c squared.
\color{blue}{a}^2 + \color{green}{b}^2 = \color{orange}{c}^2Each symbol is a length. Squaring a length gives the area of the square built on that side, matching our three coloured squares. Now a worked example. Take the legs three and four, and find the hypotenuse. Start by substituting.
\color{blue}{3}^2 + \color{green}{4}^2 = \color{orange}{c}^2Square each leg separately, giving nine plus sixteen.
\color{blue}{9} + \color{green}{16} = \color{orange}{c}^2Add them, so c squared equals twenty-five.
25 = \color{orange}{c}^2Take the positive square root, because a length cannot be negative. So c equals five.
\color{orange}{c} = \sqrt{25} = 5Now the real why. Take a big square of side a plus b and place four copies of our triangle inside it. The four triangles leave a tilted square hole. Its side is the hypotenuse, so its area is c squared.
(a+b)^2 = 4\cdot\tfrac{1}{2}ab + \color{orange}{c}^2Now slide the same four triangles into the corners. The leftover space becomes two squares, one of area a squared and one of area b squared. The outer square and the four triangles never changed, so the leftover area is the same. Therefore c squared equals a squared plus b squared.
\color{orange}{c}^2 = \color{blue}{a}^2 + \color{green}{b}^2A subtle point: the proof uses the right angle, because only then do the four triangles fit the corners with no gaps. Without it, the equation fails. In coordinates, the distance between two points is the hypotenuse of a triangle. So distance equals the square root of the sum of squared differences.
- Needs a right angle, flat geometry
d = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}Common mistake: thinking a squared plus b squared equals c squared works for any triangle. It does not, because it needs a right angle. In a non-right triangle, the law of cosines adds a correction term, c squared equals a squared plus b squared minus two a b cosine of the angle between them. At ninety degrees, cosine is zero.
c^2 = a^2 + b^2 - 2ab\cos CReal life: a ladder leans against a wall, with its foot three metres from the wall and its top four metres up. The ladder is the hypotenuse, so it is five metres long. The same idea gives diagonal braces and shortest straight-line distances on maps. The idea generalises: in three dimensions, a box diagonal has length the square root of a squared plus b squared plus c squared.
d = \sqrt{a^2+b^2+c^2}Now test your understanding with a triangle you have not seen before. To sum up: the squares on the two legs always have the same total area as the square on the hypotenuse, but only when the angle between the legs is exactly ninety degrees.
- Right angle: a² + b² = c²
Quick quiz
Q1. A triangle has sides 5, 6 and 8. Is it a right triangle?
Having a longest side does not make it right-angled; the squares must actually satisfy a squared plus b squared equals c squared.
