Question

Harry Potter bought a triangular piece of land of area $150m^2$. Harry took a piece of rope and measured the two sides of the plot and found the largest side to be $50m$ long and other side to be $10m$ long. What is the exact length of the third side?

• A. $40\sqrt{3}m$
• B. $30\sqrt{2}m$
• C. $\sqrt{1560}m$
• D. $24\sqrt{2}m$

Answer 1

The correct option is B $30\sqrt{2}m$

Let third side be $a$, then

$S=\frac{a+b+c}{2}=\frac{a+50+10}{2}=\frac{a+60}{2}$

Area of field $=\sqrt{S(S-a)(S-b)(S-c)}\Rightarrow 150=\sqrt{\left(\frac{a+60}{2}\right)(\frac{a+60}{2}-a)(\frac{a+60}{2}-50)(\frac{a+60}{2}-10)}\Rightarrow {\left(150\right)}^2=\left(\frac{a+60}{2}\right)\left(\frac{60-a}{2}\right)\left(\frac{a-40}{2}\right)\left(\frac{a+40}{2}\right)\Rightarrow 150\times 150\times 16=({60}^2-a^2)(a^2-{40}^2)\Rightarrow 150\times 150\times 16=3600a^2-5760000-a^4+1600a^2\Rightarrow a^4-5200a^2+5760000+360000=0\Rightarrow a^4-5200a^2+6120000=0a=30\sqrt{2}$

Hence, third side $=30\sqrt{2}m$