Question

The sides of triangle ABC satisfy the relations $a+b-c=2$ and $2ab-c^2=4,$ then the square of the area of triangle is __________ .

• A. $3$
• B. $4$
• C. $5$
• D. None of these

Answer 1

The correct option is A $3$

Given $a+b-c=2$ and $2ab-c^2=4$

Using these two $2ab-4=c^2=(a+b-2{)}^2$

$\Rightarrow 2ab-4=a^2+b^2+2ab-4a-4b+4\Rightarrow a^2+b^2-4a-4b+8=0$

$\Rightarrow (a-2{)}^2+(b-2{)}^2=0\Rightarrow a=b=2\Rightarrow c=2$

Thus triangle is equilateral. Hence ${\Delta }^2={(\frac{\sqrt{3}}{4}.(2{)}^2)}^2=3$