In the given figure, the circle with centre O has radius IO cm. The radius of the circle with centre P is x . STR is a common tangent to the two circles at points R and S as shown in the figure. RT= 16 cm and TS = 24 cm. What is the value of x (in cm)?

15

As we can see, Line RS and Line OP intersect each other

= Therefore, \(\angle\)RTO = \(\angle\)STP

Let \(\angle\)RTO be y

In \(\Delta \)ROT

= tan y = \(\frac{RO}{RT}\) = \(\frac{10}{16}\)

Similarly,

In \(\Delta \)STP

= tan y = \(\frac{SP}{ST}\) = \(\frac{x}{24}\)

As both the angle are equal, so we get

= \(\frac{10}{16}\) = \(\frac{x}{24}\)

= x = \(\frac{10 \;×\; 24}{16}\) = 15 cm

Therefore, the radius of circle of center P is 15 cm.