Question

The centres of two circles of radii $3cm$ and $2cm$ are $8cm$ apart. Find the length of the common tangent.

• A. 39
• B. $\sqrt{39}$
• C. $64$
• D. $25$

Answer 1

The correct option is

B

$\sqrt{39}$

In the figure, MN is the common tangent. A and B are the centres of the two circles with radii $3cm$ and $2cm$ respectively.

$AB=8cm,AM=3cm$ and $BN=2cm.$

Draw BC perpendicular to AM extended forward.
Since, BCMN is a rectangle, $CM=BN=2cm.$
Now, $AC=AM+CM=3+2=5cm.$
In $△ABC,$

$A{B}^2=A{C}^2+B{C}^2$

$\Rightarrow B{C}^2=A{B}^2-A{C}^2\Rightarrow B{C}^2=8^2-5^2\Rightarrow B{C}^2=39$
$\Rightarrow BC=\sqrt{39}$