When two chords of a circle meet at a point outside the circle, they form two segments each. The relationship between the lengths of these segments can be derived from a theorem called the Power of a Point Theorem. According to this theorem:

For two chords $AB$ and $CD$ intersecting at a point $P$ outside the circle, the following relationship holds:

$PA \cdot PB = PC \cdot PD$

Explanation:

• $PA$ and $PB$ are the segments of the chord $AB$, measured from point $P$ to the points where the chord intersects the circle.
• $PC$ and $PD$ are the segments of the chord $CD$, measured from point $P$ to the points where the chord intersects the circle.

This means that the product of the lengths of the segments of one chord equals the product of the lengths of the segments of the other chord.

Example:

If $PA = 3$, $PB = 4$, $PC = 2$, and $PD = x$, then the equation becomes: $3 \cdot 4 = 2 \cdot x$ $12 = 2x$ $x = 6$

So, the length of the segment $PD$ is 6.

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Would you like further details on this theorem or its applications?

Related Questions:

1. What is the Power of a Point Theorem and how is it derived?
2. How do you calculate the length of a chord in a circle?
3. What happens if the point $P$ lies inside the circle instead of outside?
4. How does this theorem apply when the point $P$ is on the circle itself?
5. Can the Power of a Point Theorem be used to solve real-world problems? How?

Tip:

Always remember that the Power of a Point Theorem can be a powerful tool not just in pure geometry, but also in solving problems involving circles in real-life situations, such as optics or engineering.