Two circles of radius $13CM \& 17 cm$ intersect at  $A$ and  $B$. What is the distance between the centres of the circles if  $AB=24 cm$?

 

Concept: A chord is a line segment that connects any two points on a circle's circumference. The perpendicular from the centre of a circle bisects the chord.

As since line joining the centres of the circles bisects the common chord, 

$AC=BC=12cm$.

In $∆AOC,\angle OCA= {90}^0$

By Pythagoras theorem, 

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

$\Rightarrow {15}^2=O{C}^2+{12}^2$

$O{C}^2=225-144$

$\Rightarrow OC=9 Cm$

Similarly, In 

$∆AOC, \angle O^{\text{'}}CA$

By Pythagoras theorem, 

$O^{\text{'}}{A}^2=O^{\text{'}}{C}^2+A{C}^2$

$\Rightarrow {13}^2=O^{\text{'}}{C}^2+{12}^2$

$\Rightarrow O{C}^2=169-144$

$\Rightarrow OC=5 cm$

Now, 

$O{O}^{\text{'}}=OC+O^{\text{'}}C$

$\Rightarrow O{O}^{\text{'}}=9+5=14 cm$

Hence, the distance between the centres of the circles is 14 cm.