Triangular numbers to altitudes

Use the formula that the nth triangular number is equal to $\frac{n(n+1)}{2}$ to determine that the consecutive terms are the 14th, 15th, and 16th, or $105$ , $120$ , and $136$ . Setting the 3 given expressions equal to their respective terms and using the fact that the angles of a triangle ( $a+b+c$ ) equals $180$ , it can be derived that $a=45$ , $b=60$ , and $c=75$ . Using the Law of Sines, we can find the ratio of sides. Finally, using the fact that the area of a triangle equals the base times the height divided by $2$ , the ratio of altitudes is $p=(\sqrt{3}-1) : 1 : q=\frac{(\sqrt{6})}{3}$ ). As a result, $pq=\frac{\sqrt{2}(3-\sqrt{3})}{3}$ ), and since the square root of integer $k$ is $k$ to the $\frac{(1)}{2}$ power, $0.5*3*2=3$ , which is our final answer.