Circles

There’s probably more than one way to solve this. Here’s how I would go about it.

From the intersecting chords theorem, you know that CPxDP = 4x10. You also know that CP+DP=13 (=CD). You now have two equations involving CP and DP.

Combining the two equations gives you a quadratic expression which when solved gives you CP= 5 and DP=8.

The distance from the midpoint of CD to P can be found to be 1.5.

The distance from the midpoint of AB to P can be found to be 3.0.

The distance of P to the center of the circle is the same as the diagonal of a rectangle of sides 1.5 and 3.0 which can be evaluated from the Pythagorean theorem (approximately 3.354).

quora

Let $CP = x$ and $DP = y$ and let the center of the circle be o.

We have the following system from the intersecting chord theorem: 

$xy = 40 \ \ \ \ \ \ \ \ \ \ \ \ \ \ (i)$

$x + y = 13 \ \ \ \ \ \ \ \ \ (ii)$  

Solving for x and y, we find that $x = 8$ and $y = 5$.

P is 3 units away from the midpoint of AB and 1.5 units away from the midpoint of CD.

This means that the distance is $\sqrt{3^2 + 1.5^2} = \sqrt{11.25} = \color{brown}\boxed{3 \sqrt 5 \over 2}$

Here's a graph: 

Very nice , guys   !!!!!