Question

Euclid has a triangle in mind. Its longest side has length $20$ and another of its sides has length $10$. Its area is $80$. What is the exact length of its third side?

• A. $\sqrt{240}$
• B. $\sqrt{260}$
• C. $\sqrt{280}$
• D. $\sqrt{300}$

Answer 1

The correct option is B $\sqrt{260}$

Here, In $△ABC,$

$AB=10,BC=20$

Let $AC=x,BD=a$ and $DC=20-a$

Area of of triangle $ABC=\frac{1}{2}\times BC\times AD$

$\Rightarrow$ $80=\frac{1}{2}\times 20\times h$

$\Rightarrow$ $80=10\times h$

$\Rightarrow$ $h=8$

In $△ABD$

$\Rightarrow$ $(AB{)}^2=(AD{)}^2+(BD{)}^2$

$\Rightarrow$ $(10{)}^2=(8{)}^2+a^2$

$\Rightarrow$ $100=64+a^2$

$\Rightarrow$ $a^2=36$

$\Rightarrow$ $a=6$

$\Rightarrow$ $DC=20-a=20-6=14$

In $△ADC,$

$\Rightarrow$ $(AC{)}^2=(AD{)}^2+(DC{)}^2$

$\Rightarrow$ $x^2=(8{)}^2+(14{)}^2$

$\Rightarrow$ $x^2=64+196$

$\Rightarrow$ $x=\sqrt{260}$

$\therefore$ Measure of required side is $\sqrt{260}.$