# How long is the third side of a right-angled triangle, if the hypotenuse is 13cm and the shortest side is 5cm?

Mar 9, 2017

$b = 12$

#### Explanation:

I think this is more a case of pythagoras' theorem,

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

${b}^{2} = {13}^{2} - {\left(- 5\right)}^{2}$
${b}^{2} = 169 - 25$
${b}^{2} = 144$
$b = \sqrt{144}$
$b = 12$

The missing side is $12$

Apr 25, 2018

${5}^{2} + {12}^{2} = {13}^{2}$ is a Pythagorean Triple all serious math students should recognize, and immediately answer $12$ cm to questions like this.

#### Explanation:

If you're going to be doing math, one of things you can do to really give yourself a boost is to memorize the relatively few facts that math teachers use over and over when they make up problems. For trig, mostly all you need to know are the trig functions of ${30}^{\circ} ,$ ${45}^{\circ}$ and ${60}^{\circ}$ and a few facts about supplementary and complementary angles.

It also helps to know the first few rows of some tables, like the table of Pythagorean Triples, ${a}^{2} + {b}^{2} = {c}^{2}$.

Here's one list .

${3}^{2} + {4}^{2} = {5}^{2}$
${6}^{2} + {8}^{2} = {10}^{2} \quad \quad \quad$ [3 - 4 - 5]
${5}^{2} + {12}^{2} = {13}^{2}$
${9}^{2} + {12}^{2} = {15}^{2} \quad \quad \quad$ [3 - 4 - 5]
${8}^{2} + {15}^{2} = {17}^{2}$
${12}^{2} + {16}^{2} = {20}^{2} \quad \quad \quad$ [3 - 4 - 5]
$\setminus {7}^{2} + {24}^{2} = {25}^{2}$
${15}^{2} + {20}^{2} = {25}^{2} \quad \quad \quad$ [3 - 4 - 5]

Some of these are primitive (no common factors) and some are multiples of a primitive triple, as indicated. 99% of the time when you see a Pythagorean Triple in a math question it will be one of these. You'll be giving yourself a big hint if you can recognize them when they appear.