Question

The equations of the three sides of a triangle are $x=2,y+1=0$, and $x+2y=4$.

The coordinates of the circumcenter of the triangle is

• A. $\left(2,-1\right)$
• B. $\left(0,4\right)$
• C. $\left(-1,2\right)$
• D. $\left(4,0\right)$

Answer 1

The correct option is D

$\left(4,0\right)$

Circumcentre of the triangle:

The given sides of the triangle are $x=2,y+1=0$, and $x+2y=4$.

The line $x=2$, cuts $x$-axis at $\left(2,0\right)$.

The line $y+1=0$, cuts $y$-axis at $\left(0,-1\right)$.

The line $x+2y=4$, intersects the line $x=2$ in the point $\left(2,1\right)$

Let $P=\left(2,1\right)$

Also, the line $x+2y=4$ intersects the line $y+1=0$, in the point $\left(6,-1\right)$

Let $A=\left(6,-1\right)$

Thus, the circumcentre

$$\begin{gathered} =\left(\frac{2+6}{2},\frac{1-1}{2}\right) \\ =\left(4,0\right) \end{gathered}$$

Hence, the coordinates of the circumcentre of the triangle is $\left(4,0\right)$ and therefore option $\left(D\right)$ is the correct answer.