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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.

numbermaniac
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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.

Trogdor
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  • But how would I solve for angle C without a calculator? If $200=2640sinC$, then $5/26=sinC$. – numbermaniac Mar 11 '15 at 07:47
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    You have $\sin C$, but you do not need $C$ to find $\cos C$. Simply draw up a triangle and use Pythagoras' Theorem from there. – Trogdor Mar 11 '15 at 07:49
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    I think the Pythagorus step should be included in the answer itself, for completeness. – IanF1 Mar 11 '15 at 08:00