Question

The circumcentre of the triangle with vertices $(8,6),(8,-2)$ and $(2,-2)$ is at the point

• A. $(2,-1)$
• B. $(1,-2)$
• C. $(5,2)$
• D. $(2,5)$
• E. $(4,5)$

Answer 1

The correct option is C

$(5,2)$

Explanation of the correct option:

Finding the circumcentre of the triangle:

Step 1: Equating the circumradius of triangle to find the value of $y$

Let $A(8,6),B(8,-2)$ and $C(2,-2)$ be the vertices of a triangle.

Let $P(x,y)$ be circumcentre of the triangle.

Then, $\mathrm{PA}=\mathrm{PB}=\mathrm{PC}$

Now, ${\mathrm{PA}}^2={\mathrm{PB}}^2$.

$$\begin{gathered} \therefore {(x-8)}^2+{(y-6)}^2={(x-8)}^2+{(y+2)}^2 \\ \Rightarrow {(y-6)}^2={(y+2)}^2 \\ \Rightarrow y^2-12y+36=y^2+4y+4 \\ \Rightarrow 16y=32 \\ \Rightarrow y=2 \end{gathered}$$

Step 2: Equating the circumradius of the triangle to find the value of $x$

Similarly, ${\mathrm{PB}}^2={\mathrm{PC}}^2$.

$$\begin{gathered} \therefore {(x-8)}^2+{(y+2)}^2={(x-2)}^2+{(y+2)}^2 \\ \Rightarrow {(x-8)}^2={(x-2)}^2 \\ \Rightarrow x^2-16x+64=x^2-4x+4 \\ \Rightarrow 12x=60 \\ \Rightarrow x=5 \end{gathered}$$

Hence, the coordinates of the circumcenter of a triangle is $(5,2)$.

Therefore, option (C) is the correct answer.