# An obtuse angled triangle has two of its sides 9 and 13. What is the range for the third side?

Nov 18, 2016

Either $4 < k < \sqrt{88}$

or $\sqrt{250} < k < 22$

#### Explanation:

In an obtuse angled triangle, square on the largest side is greater than sum of the squares on other two sides. However, the largest side is always less than sum of other two sides.

Now, there are two possibilities here.

(1) - Side with length $13$ is longest

In such a case, we will have ${13}^{2} > {9}^{2} + {k}^{2}$

or ${k}^{2} < 169 - 81 = 88$ and $k < \sqrt{88}$

and $13 < k + 9$ i.e. $k > 4$

i.e. $4 < k < \sqrt{88}$

(2) - Side with length $k$ is longest

In this case, we will have ${k}^{2} > {9}^{2} + {13}^{2}$

or ${k}^{2} > 81 + 169 = 250$ and $k > \sqrt{250}$

and $k < 13 + 9$ i.e. $k < 22$

i.e. $\sqrt{250} < k < 22$