given a circle with radius r two perpendicular chords intersect at point P inside the circle, dividing the circle into 4 segments. The lengths of the chords are a and b respectively. Find the product of the lengths of the line segments formed by the intersection of chords.

Let's analyze the problem step by step.

Given:
- A circle with radius \( r \).
- Two perpendicular chords intersect at point \( P \) inside the circle.
- The lengths of the chords are \( a \) and \( b \).

Let's denote the line segments created by the intersection of the chords as follows:
- Let the chord of length \( a \) be divided into two segments \( x \) and \( a-x \).
- Let the chord of length \( b \) be divided into two segments \( y \) and \( b-y \).

The goal is to find the product of the lengths of these segments: \( x \cdot (a-x) \cdot y \cdot (b-y) \).

### Step 1: Use the Power of a Point Theorem
By the Power of a Point Theorem (or Intersecting Chords Theorem), for any point \( P \) inside the circle, the products of the lengths of the segments of each chord are equal. That is:
\[
x \cdot (a - x) = y \cdot (b - y)
\]

### Step 2: Calculate the Required Product
We need to find \( x \cdot (a-x) \cdot y \cdot (b-y) \).

Let \( x \cdot (a-x) = p \) and \( y \cdot (b-y) = q \).

Since \( p = q \), the product is:
\[
x \cdot (a-x) \cdot y \cdot (b-y) = p \cdot q = p^2
\]

###

Explore how to calculate the product of line segments formed by the intersection of two perpendicular chords inside a circle. This problem involves the Power of a Point Theorem and is suitable for advanced high school students.

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