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  4. Let the points of intersections of the lines x-y+1=0, x-2 y+3=0 and 2 x-5 y+11=0 are the mid points of the sides of a triangle ABC. Then the area of the triangle ABC is

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Q. Let the points of intersections of the lines $x-y+1=0$, $x-2 y+3=0$ and $2 x-5 y+11=0$ are the mid points of the sides of a triangle $ABC$. Then the area of the triangle $ABC$ is ___

JEE MainJEE Main 2021Straight Lines

Solution:

intersection point of give lines are $(1,2),(7,5)$ $(2,3)$
image
$\Delta=\frac{1}{2}\begin{vmatrix}1 & 2 & 1 \\ 7 & 5 & 1 \\ 2 & 3 & 1\end{vmatrix}$
$=\frac{1}{2}[1(5-3)-2(7-2)+1(21-10)]$
$=\frac{1}{2}[2-10+11]$
$\Delta DEF =\frac{1}{2}(3)=\frac{3}{2}$
$\Delta ABC =4 \Delta DEF =4\left(\frac{3}{2}\right)=6$