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  4. If the vertex opposite to the hypotenuse of a right-angled triangle lies on the straight line 2x+y-10=0 and the two other vertices are (2 , - 3) and (4 , 1) , then the area of the triangle (in sq. units) is equal to

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Q. If the vertex opposite to the hypotenuse of a right-angled triangle lies on the straight line $2x+y-10=0$ and the two other vertices are $\left(2 , - 3\right)$ and $\left(4 , 1\right)$ , then the area of the triangle (in sq. units) is equal to

NTA AbhyasNTA Abhyas 2020Straight Lines

Solution:

Solution
$m_{1}m_{2}=-1\Rightarrow \frac{9 - 2 a}{a - 4}\times \frac{13 - 2 a}{a - 2}=-1$
$\Rightarrow 5a^{2}-50a+125=0$
So, $a=5$ and $B$ is $\left(5 , 0\right)$
Hence, required area $=\frac{1}{2}AB\times AC=\frac{1}{2}\sqrt{2}\times 3\sqrt{2}=3$ sq. units