Pythagorean theorem

It is a statement about areas

Draw a real square on each side of the triangle. The theorem says the two smaller squares hold exactly as much area as the big one.

$$\underbrace{9}_{3\times3} + \underbrace{16}_{4\times4} = \underbrace{25}_{5\times5}$$

That is a claim about area, not lengths — $3 + 4 \ne 5$, and was never meant to. Only after squaring do the sides fit together.

Which side is c

Always the hypotenuse: opposite the right angle, and always the longest side.

The practical test is size. If one known side is longer than the other, it is $c$ and the calculation subtracts. If both known sides are shorter than the missing one, they are $a$ and $b$ and the calculation adds.

You know	You want	Do
both legs	the hypotenuse	add the squares
hypotenuse and a leg	the other leg	subtract the squares

Rearranged

$$c = \sqrt{a^2 + b^2} \qquad a = \sqrt{c^2 - b^2}$$

Do not forget the square root. The theorem gives the square of the side, and one step always remains.

Triples worth recognising

Triple	Check
$3, 4, 5$	$9 + 16 = 25$
$5, 12, 13$	$25 + 144 = 169$
$8, 15, 17$	$64 + 225 = 289$
$7, 24, 25$	$49 + 576 = 625$

Any multiple of a triple is another triple — $6, 8, 10$ and $9, 12, 15$ are the $3, 4, 5$ triangle scaled. Scaling multiplies every side alike, so the relationship survives.

The converse is true too

If a triangle’s sides satisfy $a^2 + b^2 = c^2$, the triangle must be right-angled. That is how a builder checks a corner: measure $3$ along one wall, $4$ along the other, and confirm the diagonal is exactly $5$.

The signature error

Adding when you should subtract gives a leg longer than the hypotenuse, which is impossible for a right triangle. The answer fails on inspection, before any arithmetic is rechecked. See the Pythagorean theorem.