# How do you use the Pythagorean Theorem to determine if the three numbers could be the measures of the sides of a right triangle: 6, 12, 18?

Feb 29, 2016

Check the similar triangle with sides $1$, $2$, $3$ against Pythagoras formula to find that this is not a right angled triangle.

#### Explanation:

Three positive numbers can be the measures of the sides of a right triangle if and only if taken in ascending order they satisfy:

${a}^{2} + {b}^{2} = {c}^{2}$

Also a triangle is a right triangle if and only if any similar triangle is a right triangle. So you can multiply or divide $a$, $b$ and $c$ by any non-zero number before applying the test.

In our example, all of the sides are divisible by $6$ so let us assign:

$a = \frac{6}{6} = 1$

$b = \frac{12}{6} = 2$

$c = \frac{18}{6} = 3$

We find ${a}^{2} + {b}^{2} = {1}^{2} + {2}^{2} = 1 + 4 = 5 \ne 9 = {3}^{2} = {c}^{2}$

In fact these side lengths only form a degenerate 'triangle' of zero area, with interior angles $0$, $0$ and $\pi$