Question

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

Answer 1

Assuming the missing side, c, is the hypotenuse
by the Pythagorean Theorem
`c^2 = a^2+b^2`
`= 5^2 +12^2`
`= 169`
`rarr 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`
`rarr c = sqrt(119)`

Answer 2

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=2k+3`

`b=(a^2-1)/2=2k^2+6k+4`

`c=(a^2+1)/2=2k^2+6k+5`

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