Question

If $AB=12cm,BC=16cm$, and $AB$ is perpendicular to $BC$, then the radius of the circle passing through the points$A,B,$ and $C$ is

• A. 6 cm
• B. 8 cm
• C. 10 cm
• D. 12 cm

Answer 1

The correct option is C

10 cm

Explanation for correct option

Step 1: Construction

• $AB=12cm$
• $BC=16cm$
• $AB\perp BC$

Hence,

$AC$ is the diameter of the circle passing through the points $A,B$ and$C$

Step 2: Apply the Pythagoras theorem.

Now, as per the diagram $ABC$ is a right-angled triangle.

On applying Pythagoras theorem:

${\left(AC\right)}^2={\left(CB\right)}^2+{\left(AB\right)}^2$

$\Rightarrow {\left(AC\right)}^2={\left(16\right)}^2+{\left(12\right)}^2$

$\Rightarrow {\left(AC\right)}^2=256+144$

$\Rightarrow {\left(AC\right)}^2=400$

Step 3: To find the radius.

Take square root on LHS and RHS to calculate the diameter and radius

$\sqrt{{\left(AC\right)}^2}={\left(400\right)}^2$

$\left(AC\right)=20$

Diameter of the circle = $20cm$

Thus, $radiusofthecircle=\frac{Diameter}{2}$

$$\begin{gathered} =\frac{20}{2} \\ =10cm \end{gathered}$$

Thus, the Radius of the circle =$10cm$.

Hence, option(C) is the correct answer.