# How to Calculate Segment Lengths in Circles Using the Power Theorem

In the vast realm of geometry, circles hold a unique fascination, primarily due to their perfect symmetry and endless periphery. One of the intriguing aspects of circle geometry is the study of segment lengths within these closed curves. Segments, which are areas bound by chords and arcs, vary in length depending on their position and the angles subtended by their bounding chords. The lengths of these segments can reveal a wealth of information about the circle and its associated properties. As we delve deeper into the subject, we'll uncover the mathematical intricacies behind these lengths and explore how they interplay with other geometric elements of the circle, laying the groundwork for further investigations into the more advanced concept of the power theorem.

## Step-by-step Guide: Segment Lengths in Circle

**1. Chord-Chord Power Theorem**:

When two chords intersect inside a circle, the product of the lengths of the segments of one chord equals the product of the lengths of the segments of the other chord.

Mathematically, if \(AB\) and \(CD\) are intersecting chords, then:

\( AE \times EB = CE \times ED \)

**2. Tangent-Secant Power Theorem**:

When a tangent and a secant intersect outside a circle, the square of the length of the tangent segment equals the product of the lengths of the secant’s entire length and its external segment.

Mathematically, if \(PA\) is a tangent and \(PBC\) is a secant, then:

\( PA^2 = PB \times PC \)

**3. Secant-Secant Power Theorem**:

When two secants intersect outside a circle, the product of the lengths of one secant and its external segment equals the product of the lengths of the other secant and its external segment.

Mathematically, if \(PAB\) and \(PCD\) are intersecting secants, then:

\( PA \times PB = PC \times PD \)

### Examples

**Example 1**:

Two intersecting chords in a circle, \(AB\) and \(CD\), are divided into segments of \(3 \text{ cm}\), \(4 \text{ cm}\), \(2 \text{ cm}\), and \(x\) respectively. Using the Chord-Chord Power Theorem, determine the value of \(x\).

**Solution**:

According to the Chord-Chord Power Theorem:

\( AE \times EB = CE \times ED \)

Given: \( AE = 3 \text{ cm} \) and \( EB = 4 \text{ cm} \)

\( 3 \times 4 = 2 \times x \)

\( x = 6 \text{ cm} \)

**Example 2**:

A circle has a tangent \(PA\) of length \(5 \text{ cm}\) and a secant \(PBC\) with segment \(PB\) measuring \(3 \text{ cm}\) and \(PC\) measuring \(7 \text{ cm}\). Validate the Tangent-Secant Power Theorem with the given data.

**Solution**:

According to the Tangent-Secant Power Theorem:

\( PA^2 = PB \times PC \)

Given: \( PA = 5 \text{ cm} \), \( PB = 3 \text{ cm} \), \( PC = 7 \text{ cm} \)

\( 5^2 = 3 \times 7 \)

This holds false since \(25 \neq 21\).

**Example 3**:

Two secants, \(PAB\) and \(PCD\), intersect outside a circle. The given measurements are: \(PA = 2 \text{ cm}\), \(AB = 4 \text{ cm}\), \(PC = 3 \text{ cm}\), and \(CD = y\). Use the Secant-Secant Power Theorem to determine the value of \(y\).

**Solution**:

According to the Secant-Secant Power Theorem:

\( PA \times PB = PC \times PD \)

Given: \( PB = PA + AB = 2+4 = 6 \text{ cm} \), \( PD = 3 + y \text{ cm} \)

\( 2 \times 6 = 3 \times (3+y) \)

On solving, \( y = 1 \text{ cm} \).

### Practice Questions:

- Two intersecting chords in a circle have lengths of segments as \(2 \text{ cm}\), \(8 \text{ cm}\), \(3 \text{ cm}\) and \(x\). Find \(x\).
- A tangent of length \(6 \text{ cm}\) intersects with a secant of full length \(10 \text{ cm}\) and what is the external segment?

**Answers**:

- \( x = 5.33 \text{ cm} \)
- External segment \( = 3.6 \text{ cm} \)

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