Math Core

The Pythagorean theorem starts with a right triangle and tells you something about its sides. But what if you only know the three side lengths? Can you tell whether the triangle has a right angle without measuring it? Yes: the theorem also works in reverse.

Turning the theorem around

The Pythagorean theorem says: if a triangle is a right triangle, then $a^2 + b^2 = c^2$ . Switching the "if" part and the "then" part gives a new statement, called the converse. For this theorem, the converse is also true.

The converse of the Pythagorean theorem

If the side lengths of a triangle satisfy $a^2 + b^2 = c^2$ , where $c$ is the longest side, then the triangle is a right triangle. The right angle is across from the longest side $c$ .

If $a^2 + b^2 \ne c^2$ , the triangle is not a right triangle.

Because 5² + 12² = 13², a triangle with sides 5, 12 and 13 must have a right angle, across from the side of length 13.

Why the converse is true

Suppose a triangle has sides $a$ , $b$ and $c$ with $a^2 + b^2 = c^2$ . Now draw a second triangle that you know is a right triangle, with legs $a$ and $b$ . By the Pythagorean theorem, its hypotenuse is $\sqrt{a^2 + b^2}$ , which equals $\sqrt{c^2} = c$ .

So the two triangles have exactly the same three side lengths: $a$ , $b$ and $c$ . Triangles with the same three side lengths are congruent, so one fits exactly on top of the other. That means the first triangle has a right angle too.

Testing a triangle

To decide whether three lengths make a right triangle:

Find the longest side. Call it $c$ .
Square the two shorter sides and add them.
Compare the sum with $c^2$ . Equal means right triangle. Not equal means not a right triangle.

Worked example: Sides 9, 12 and 15

Is a triangle with sides $9$ , $12$ and $15$ a right triangle?

The longest side is $15$ .

$9^2 + 12^2 = 81 + 144 = 225 \qquad 15^2 = 225$

The two results are equal, so yes, it is a right triangle.

Worked example: Sides 6, 7 and 9

Is a triangle with sides $6$ , $7$ and $9$ a right triangle?

The longest side is $9$ .

$6^2 + 7^2 = 36 + 49 = 85 \qquad 9^2 = 81$

Since $85 \ne 81$ , no, it is not a right triangle.

Worked example: Sides in a mixed-up order

A triangle has sides $26$ , $10$ and $24$ . Is it a right triangle? If so, which side is across from the right angle?

The sides aren't listed in order, so find the longest first: it's $26$ .

$10^2 + 24^2 = 100 + 576 = 676 \qquad 26^2 = 676$

They match, so it is a right triangle. The right angle is across from the side of length $26$ .

Common mistake

Always put the longest side in the $c$ spot. With sides $26$ , $10$ and $24$ , checking $26^2 + 10^2 = 24^2$ gives $776 = 576$ , which is false, and you would wrongly decide the triangle isn't right. The longest side never gets added; it goes alone on the other side of the equals sign.

Pythagorean triples

Three whole numbers that satisfy $a^2 + b^2 = c^2$ are called a Pythagorean triple. Some triples show up so often that they are worth knowing on sight:

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

Multiply every number in a triple by the same amount and you get another triple. For example, doubling $3, 4, 5$ gives $6, 8, 10$ , and tripling it gives $9, 12, 15$ . (That's why $9$ , $12$ , $15$ worked in the first example.)

Checking corners in real life

Builders use the converse to make sure corners are square. To check a corner, they measure $3$ feet along one wall and $4$ feet along the other, and mark both spots. If the distance between the marks is exactly $5$ feet, the corner is a right angle.

Worked example: Is the patio square?

Ana is building a patio. She measures $6$ feet along one edge and $8$ feet along the other edge from the same corner. The distance between her two marks is $10$ feet. Is the corner a right angle?

$6^2 + 8^2 = 36 + 64 = 100$ and $10^2 = 100$ . They are equal, so yes, the corner is a right angle.

Tip

Before squaring, see whether the sides share a common factor. The sides $15$ , $20$ and $25$ are $5$ times $3$ , $4$ and $5$ , so they form a right triangle. No big squares needed.

Practice

Practice 1

Is a triangle with sides $8$ , $15$ and $17$ a right triangle?

Practice 2

Is a triangle with sides $5$ , $6$ and $8$ a right triangle?

Practice 3

Which set of side lengths forms a right triangle?

Practice 4

A triangle has sides $6$ , $10$ and $8$ . It is a right triangle. The right angle is across from which side?

Practice 5

A triangle has sides $12$ and $16$ . What length should the longest side be for the triangle to be a right triangle?

Enter a number. Fractions like 3/4 and sqrt(2) are OK.

Practice 6

A carpenter builds a rectangular frame with sides $30$ inches and $40$ inches. To check that the corners are square, she measures the diagonal. How long should the diagonal be, in inches, if the corners are right angles?

Enter a number. Fractions like 3/4 and sqrt(2) are OK.

Practice 7

Is a triangle with sides $\sqrt{7}$ , $3$ and $4$ a right triangle?

Practice 8

How many of these sets of side lengths form right triangles?

$9, 40, 41 \qquad 11, 60, 61 \qquad 6, 8, 11 \qquad 12, 35, 37$

Enter a number. Fractions like 3/4 and sqrt(2) are OK.