Chord Theorem and Tangent

A chord is a line segment joining two points on a circle. When two chords intersect inside a circle, an important relation is formed.

1. Two chords intersect at a point inside the circle

If two chords intersect at a point inside the circle, then the product of the segments of one chord is equal to the product of the segments of the other chord.

AP × PB = CP × PD

Diagram

Example

If

$AP = 3, PB = 4,$ and $CP = 2$

Using the chord theorem:

Key Point

• This theorem works only when chords intersect inside the circle.
• Frequently asked in SSC, Banking, and Railway exams.

2. Intersecting Chords Theorem

The Power of a Point theorem explains relationships between chords, secants, and tangents drawn from a point to a circle.

If

$AP = 3, PB = 4,$ and $CP = 2$

Using the chord theorem:

This theorem is frequently used in geometry problems where chord lengths are given.

Trick: there will be P in every line segment

$$A\mathbf{P} \times \mathbf{P}B = C\mathbf{P} \times \mathbf{P}D$$

3. Power of a Point

The power of a point theorem states that for a point relative to a circle, the product of the lengths of segments of a secant line from the point is constant and equal to the square of the tangent length from that point to the circle.

Example:

From a point (P) outside a circle, a tangent (PT) and a secant (PAB) are drawn.

If

Find the length of the tangent (PT).

Solution

According to the {Power of a Point Theorem:

Substitute the values: