Why do this problem?

The problem calls for simple geometrical reasoning (nothing more complicated than similar triangles) and leads to an obvious generalisation. Finding the solution is the beginning of the next stage:

how can I explain my method?
is this the best method?
what if...can I generalise this result?

Key questions

What do you know about the tangent to a circle and its radius?

What do you know about the perpendicular from the centre of a circle to a chord?

Can you spot any similar triangles?

If two sides of a right-angled triangle are in the ratio 3:5 what can we say about the third side?