Question

The line $3x+2y=24$ meets the y-axis at $A$ & the x-axis at $B$. The perpendicular bisector of $AB$ meets the line through $(0,-1)$ parallel to x-axis at $C$. Find the area of the $△ABC$.

• A. $91$ sq. units
• B. $5$ sq.units
• C. $182$ sq.units
• D. $10$ sq.units

Answer 1

The correct option is A $91$ sq. units

$A(0,12),B(8,0)$

$D$ the mid point of $AB$ is $(4,6)$

Equation of the perpendicular bisector is

$2x-3y=2\times 4-3\times 6=-10$

which meets the line $y=-1$ at $C(-\frac{13}{2},-1)$

Area of triangle $ABC$

$=\frac{1}{2}AB\times CD$

$=\frac{1}{2}\sqrt{64+144}+\sqrt{{(4+\frac{13}{2})}^2+{(6+1)}^2}$

$=\frac{1}{2}\times 4\sqrt{13}\times 7\sqrt{{\left(\frac{3}{2}\right)}^2+1}$

$=13\times 7=91$