Question

The circumcenter of the triangle whose vertices are $\left(0,0\right),(3,0)$and $\left(0,4\right)$is

• A. $\left(\frac{3}{2},2\right)$
• B. $\left(2,\frac{3}{2}\right)$
• C. $\left(0,0\right)$
• D. None of these

Answer 1

The correct option is A

$\left(\frac{3}{2},2\right)$

Explanation for the correct option:

To find the circumcenter of the triangle:

Let $A(0,0),B(3,0)$and $C(0,4)$be the vertex and the circumcenter be $O(h,k)$

We know that circumcenter is equidistant to vertices.

So, $OA=OB=OC$

Now, ${\left(OA\right)}^2={\left(OB\right)}^2$

$$\begin{gathered} \Rightarrow {(h-0)}^2+{(k-0)}^2={(3-h)}^2+{(0-k)}^2 \\ \Rightarrow h^2+k^2=9+h^2-6h+k^2 \\ \Rightarrow 6h=9 \\ \Rightarrow h=\frac{3}{2} \end{gathered}$$

and ${\left(OA\right)}^2={\left(OC\right)}^2$

$$\begin{gathered} \Rightarrow {(h-0)}^2+{(k-0)}^2={(0-h)}^2+{(4-k)}^2 \\ \Rightarrow h^2+k^2=h^2+16+k^2-8k \\ \Rightarrow 8k=16 \\ \Rightarrow k=2 \end{gathered}$$

Thus the coordinates of the circumcenter is $\left(h,k\right)=\left(\frac{3}{2},2\right)$

Hence, the correct option is (A).