How to Calculate Segment Lengths in Circles Using the Power Theorem

Segment lengths in a circle follow three power theorems: chord-chord, secant-secant and secant-tangent. Each says that two products of segment lengths are equal, so one unknown length can always be solved for. Below: all three stated, worked examples, and practice questions.

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 \(25 \neq 21\).

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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:

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

Answers:

1. \( x = 5.33 \text{ cm} \)
2. External segment \( = 3.6 \text{ cm} \)