How to Calculate Segment Lengths in Circles Using 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:

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 $= 3.6 \text{ cm}$