A triangle has an area of $200cm^{2}$. Two sides of this triangle measure 26 and 40 cm respectively. Find the exact value of the third side.
I used Heron's formula to solve this equation, but it involves a long calculation and then the quadratic formula for a final solution. But, somehow I got it wrong along the way and my answer was x= $2\sqrt {431}+40$ . However, the given answer is $2 \sqrt{89}$, and when I use my CAS calculator to solve the question I do get that as my answer. Is there a faster way to solve these kind of questions if I can't use a calculator? Thanks.
Use the fact that the area of a triangle is given by $A=\frac{1}{2}ab \sin C$ where $C$ is the angle between the two sides of length $a$ and $b$. You should be able to solve for $\theta$.
Once you do so, you can compute the length of the last side using the Cosine Rule.