How do I find the length between two circles that have the same tangent line?

Question

P and N are the center of the two circles with radii 50 units and 5 units respectively. TS is the common tangent to the circles at point Z and R. TNP is a straight line and the distance between P and N is 170 units.

Find the length of ZR.

I'm stuck doing this problem. Can someone teach me how do you solve this. Which theorems do you need to use to solve this problem?

Answer 1

$$\Delta TNR\sim\Delta TPZ$$

Hence, $$\frac{NT}{NR}=\frac{NT+PN}{PZ}$$ $$\frac{NT}{5}=\frac{NT+170}{50}$$ $$50NT=5NT+850$$ $$45NT=850$$ $$NT=\frac{850}{45}=\frac{170}{9}$$ $$PT=PN+NT=170+\frac{170}{9}=\frac{1700}{9}$$

Now, use the Pythagoras theorem on $\Delta TPZ$.

Answer 2

Hint. $PZ$ and $NR$ are parallel (why?). Hence the triangles $PZT$ and $NRT$ are similar: $$\frac{ZT}{PZ}=\frac{RT}{NR}\quad \mbox{and}\quad \frac{PT}{PZ}=\frac{NT}{NR}.$$