Question

Let the orthocentre and centroid of a triangle be $A\left(-3,5\right)$ and $B\left(3,3\right)$ respectively. If $C$ is the circumcentre of this triangle, then the radius of the circle having line segment $AC$ as diameter, is

• A. $3\sqrt{\frac{5}{2}}$
• B. $\frac{3\sqrt{5}}{2}$
• C. $\sqrt{10}$
• D. $2\sqrt{10}$

Answer 1

The correct option is A

$3\sqrt{\frac{5}{2}}$

Explanation for the correct option:

Finding the radius of the circle:

Step 1: Given Details

Given the coordinates of the orthocentre of a triangle $A\left(-3,5\right)$ and the coordinates of the centroid of a triangle $B\left(3,3\right)$.

Let the coordinates of the circumcentre of the triangle be $C\left(x,y\right)$.

From the Euler theorem, we know that the centroid $B$ always divides the line connecting the orthocentre $A$ and the circumcentre $C$ in the ratio $2:1$.

Step 2: Finding the coordinates of $C$,

Using the internal section formula we get

$C_{\left(x,y\right)}=\left(\frac{m{x}_2+n{x}_1}{m+n},\frac{m{y}_2+n{y}_1}{m+n}\right)$

Now finding the $C_x$

$$\begin{gathered} \Rightarrow C_x=\frac{m{x}_2+n{x}_1}{m+n} \\ \Rightarrow 3=\frac{2x+\left(-3\right)}{2+1} \\ \Rightarrow 3=\frac{2x-3}{3} \\ \Rightarrow 9=2x-3 \\ \Rightarrow x=6 \end{gathered}$$

and

$$\begin{gathered} \Rightarrow C_y=\frac{m{y}_2+n{y}_1}{m+n} \\ \Rightarrow 3=\frac{2y+1\left(5\right)}{2+1} \\ \Rightarrow 3=\frac{2y+5}{3} \\ \Rightarrow 9=2y+5 \\ \Rightarrow y=2 \end{gathered}$$

Therefore, the coordinates of $C$ is $\left(6,2\right)$.

Step 3: Determining the distance

Since the line segment $AC$ is the diameter we are using the distance formula to find the distance from $A$ to$C$.

$$\begin{gathered} AC=\sqrt{{\left(x_2-x_1\right)}^2+{\left(y_2-y_1\right)}^2} \\ AC=\sqrt{{\left(6+3\right)}^2+{\left(2-5\right)}^2} \\ AC=\sqrt{81+9} \\ AC=\sqrt{90} \end{gathered}$$

Therefore, the diameter is obtained as $\sqrt{90}$.

We know that $\text{2×radius}=\text{diameter}$

$$\begin{gathered} 2r=AC \\ r=\frac{\sqrt{90}}{2} \\ =\sqrt{\frac{90}{4}} \\ =\sqrt{\frac{9\times 10}{4}} \\ =3\sqrt{\frac{5}{2}} \end{gathered}$$

Therefore, the radius of the circle having a line segment $AC$ is $3\sqrt{\frac{5}{2}}$.

Hence, the correct answer is option (A)