Question

Assertion :Let the vertices of a $\Delta ABC$ are $A(-5,-2),B(7,6)$ and $C(5,-4)$, then coordinate of circumcentre is $(1,2)$. Reason: In a right angle triangle, mid-point of hypotenuse is the circumcentre of the triangle.

• A. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
• B. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
• C. Assertion is correct but Reason is incorrect
• D. Assertion is incorrect but Reason is correct

Answer 1

The correct option is A Both Assertion and Reason are correct and Reason is the correct explanation for Assertion

$ABC$ is a triangle with vertices $A(x_1,y_1)=(-5,-2),B(x_2,y_2)=(7,6)$ and $C(x_3,y_3)=(5,-4)$, then

$m_{AB}=\frac{y_2-y_1}{x_2-x_1}=\frac{6+2}{7+5}=\frac{2}{3}$,

$m_{BC}=\frac{y_3-y_2}{x_3-x_2}=\frac{-4-6}{5-7}=5$,

$m_{AC}=\frac{y_3-y_1}{x_3-x_1}=\frac{-4+2}{5+5}=-\frac{1}{5}$.

So, clearly $m_{BC}\times m_{AC}=-\frac{1}{5}\times 5=-1$

$\therefore \Delta ABC$ is a right angled, one with $\angle C={90}^o$.

So, the hypotenuse is $AB$.

Let the co-ordinates of the cicumcente of $\Delta ABC$ are $P(x,y)$.

Since a right triangle can be inscribed within a semicicle with

diameter $=$ the hypotenuse, the centre will be the mid point of the hypotenuse.

So, the mid point of $AB$ is $P(x,y)=(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2})$

$=(\frac{-5+7}{2},\frac{-2+6}{2})=(1,2)$.

So, the assertion as well as the reason is correct and the reason is the correct explanation for the given assertion.