Pythagoras' theorem

In a right triangle one of the angles is $\displaystyle 90^{\circ}$. Its sides have names:

• The legs – the two sides that make the right angle: $\displaystyle a$ and $\displaystyle b$.
• The hypotenuse – the side opposite the right angle: $\displaystyle c$.

On each side of a right triangle we built a square. The legs of the triangle are 3 and 4, and the hypotenuse is 5:

• The square on the leg 3 has an area of 9, and on the leg 4 the area is 16. Count the squares of the grid.
• The square on the hypotenuse has an area of $\displaystyle {5^{2}}={25}$.
• $\displaystyle {9+16}={25}$: the square on the hypotenuse has the same area as the two squares on the legs together.

Pythagoras’ theorem: in a right triangle, the sum of the squares of the legs equals the square of the hypotenuse.

• $\displaystyle a$ and $\displaystyle b$ are the legs, and $\displaystyle c$ is the hypotenuse.
• The theorem holds in every right triangle, and only in a right triangle.
• The hypotenuse is the longest side of a right triangle: $\displaystyle c^{2}$ is larger than both $\displaystyle a^{2}$ and $\displaystyle b^{2}$.

The proof below builds two equal large squares, takes four triangles like ours out of each, and compares what is left.

• The two large squares have the same side, $\displaystyle a+b$, so the same area.
• Each holds four shaded triangles: right triangles with legs $\displaystyle a$ and $\displaystyle b$, and hypotenuse $\displaystyle c$.
• In the left square, what is left is a square with side $\displaystyle c$. Its area is $\displaystyle c^{2}$.
  • It is a square: all its sides are hypotenuses, $\displaystyle c$ long. At each of its corners, its angle and the triangle’s two acute angles together make $\displaystyle 180^{\circ}$, and the two acute angles together are $\displaystyle 90^{\circ}$ – so its angle is $\displaystyle 90^{\circ}$ too.
• In the right square, what is left is two squares, with areas $\displaystyle a^{2}$ and $\displaystyle b^{2}$.
• We took the same area out of two equal areas, so what is left is equal: $\displaystyle {c^{2}}={a^{2}+b^{2}}$.

The legs of a right triangle are 6 and 8.
We find the hypotenuse $\displaystyle c$.

• By Pythagoras’ theorem: $\displaystyle {c^{2}}={6^{2}+8^{2}}={36+64}={100}$.
• $\displaystyle c$ is a length, a positive number whose square is 100. So $\displaystyle {c}={\sqrt{100}}={10}$.

In one line: $\displaystyle {c}={\sqrt{6^{2}+8^{2}}}={\sqrt{100}}={10}$.

In each part, the legs of a right triangle: 5 and 12; 9 and 12; 8 and 15; 0.6 and 0.8.
Find the hypotenuse.

In triangle $\displaystyle PQR$, the angle $\displaystyle P$ is a right angle.
Which side is the hypotenuse?
Write Pythagoras’ theorem for this triangle, with the letters of its sides.

The legs of a right triangle are 1 and 2.
Find the hypotenuse.
Between which two whole numbers is it?

Find the length of the segment $\displaystyle AB$.