# How do you find the missing side of the right triangle with legs: a = 5, b = 12?

May 24, 2015

Assuming the missing side, c, is the hypotenuse
by the Pythagorean Theorem
${c}^{2} = {a}^{2} + {b}^{2}$
$= {5}^{2} + {12}^{2}$
$= 169$
$\rightarrow c = 13$

If the missing side, c, is not the hypotenuse
then b, the longer side must be the hypotenuse, and
by the Pythagorean Theorem
${b}^{2} = {c}^{2} + {a}^{2}$
or
${c}^{2} = {b}^{2} - {a}^{2}$
$= {12}^{2} - {5}^{2}$
$= 119$
$\rightarrow c = \sqrt{119}$

May 24, 2015

Use Pythagoras theorem: "The square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the other two sides"

${a}^{2} + {b}^{2} = {5}^{2} + {12}^{2} = 25 + 144 = 169 = {13}^{2}$

So the hypotenuse of your triangle has length $13$.

You have probably encountered the 3-4-5 triangle. The 5-12-13 triangle is part of the same sequence of right angled triangles with integer length sides:

If $k$ is a non-negative integer, then

$a = 2 k + 3$

$b = \frac{{a}^{2} - 1}{2} = 2 {k}^{2} + 6 k + 4$

$c = \frac{{a}^{2} + 1}{2} = 2 {k}^{2} + 6 k + 5$

are the lengths of the sides of a right angled triangle.