No Height

Relevant wiki: Pythagorean Theorem

Since the triangle on the right is a right triangle, the triangle on the left must also be a right triangle. Using the Pythagorean Theorem, the length of the middle line (the height of the big triangle) must be $\sqrt{169-25}=12$ Using PT again, we get $\sqrt{144+256}=\boxed{20}.$

Let the top, left and right vertices be $A$ , $B$ and $C$ respectively and the leg of altitude from point $A$ to $BC$ be $D$ . We note that $\triangle ABD$ is a 5-12-13 Pythagorean triangle , therefore $AD = 12$ . $\triangle ADC$ is a 3-4-5 Pythagorean triangle, therefore, $AC = 4\times 5 = \boxed{20}$ .