Two perpendicular chords AB and CD intersect at P. If PA = 4, PB = 10, and CD = 13, calculate the length of P to the center of the circle

Guest Jul 29, 2022

#1**+6 **

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

nerdiest Jul 29, 2022

#2**0 **

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:

BuilderBoi Jul 29, 2022