Determining Chord Segment Length Using Circle Geometry

This problem requires us to find the length of a segment of a chord when two chords intersect externally. We will apply the **Power of a Point Theorem**, specifically the variant for two secants intersecting outside a circle, known as the **Intersecting Secants Theorem**. This theorem states that if two secant lines are drawn from an external point P to a circle, intersecting the circle at two points each (say, A, B and C, D), then the product of the lengths of the segments from P to the intersection points is equal for both lines.

Mathematically, the theorem is expressed as:

$$ PA \cdot PB = PC \cdot PD $$

Where PA and PB are the lengths of the segments from P to the circle along one secant, and PC and PD are the lengths from P to the circle along the other secant.

Given Information Analysis

We are provided with the following details:

• The length of chord AB is $AB = 7$ cm.
• The length of chord CD is $CD = 1$ cm.
• The distance from the external point P to point D is $PD = 5$ cm.

Our goal is to calculate the length of the segment PB.

Analyzing the Secant PCD

Consider the line segment extending from P through C and D to the circle. P is the external point. C and D are points on the circle. The chord length is $CD = 1$ cm, and $PD = 5$ cm.

There are two possible configurations for the points P, C, D along the line:

1. Configuration 1: P--C--D
   In this case, C lies between P and D. The distance $PD$ is the sum of $PC$ and $CD$. $$ PD = PC + CD $$ Substituting the given values: $$ 5 = PC + 1 $$ Solving for $PC$, we find $PC = 5 - 1 = 4$ cm. The product of the segments from P is $PC \cdot PD = 4 \times 5 = 20$.
2. Configuration 2: P--D--C
   In this case, D lies between P and C. The distance $PC$ is the sum of $PD$ and $CD$. $$ PC = PD + CD $$ Substituting the given values: $$ PC = 5 + 1 = 6 \text{ cm} $$ The product of the segments from P is $PC \cdot PD = 6 \times 5 = 30$.

Analyzing the Secant PAB

Now consider the line segment extending from P through A and B to the circle. P is the external point. A and B are points on the circle. The chord length is $AB = 7$ cm. Let the length $PB = x$. We need to find $x$.

There are two possible configurations for the points P, A, B along the line:

1. Configuration A: P--A--B
   In this case, A lies between P and B. The distance $PB$ is the sum of $PA$ and $AB$. $$ PB = PA + AB $$ $$ x = PA + 7 $$ This implies $PA = x - 7$. For this arrangement to be possible, $PB$ must be greater than $AB$, so $x > 7$. The product of the segments from P is $PA \cdot PB = (x - 7) \times x$.
2. Configuration B: P--B--A
   In this case, B lies between P and A. The distance $PA$ is the sum of $PB$ and $AB$. $$ PA = PB + AB $$ $$ PA = x + 7 $$ The product of the segments from P is $PA \cdot PB = (x + 7) \times x$.

Applying the Theorem and Solving

We equate the products calculated from the two secants using the Intersecting Secants Theorem: $PA \cdot PB = PC \cdot PD$. We examine the scenarios based on the possible products from secant PCD (20 or 30).

Evaluating Scenario 1 (Product = 20)

If $PC \cdot PD = 20$ (from Configuration 1: P--C--D):

• Matching with Configuration A (P--A--B): $(x - 7)x = 20 \implies x^2 - 7x - 20 = 0$. The solutions are $x = \frac{7 \pm \sqrt{129}}{2}$. These are not integer values matching the options.
• Matching with Configuration B (P--B--A): $(x + 7)x = 20 \implies x^2 + 7x - 20 = 0$. The solutions are $x = \frac{-7 \pm \sqrt{129}}{2}$. These are not positive integer values.

Since neither arrangement yields a valid answer from the options when the product is 20, we consider the other scenario.

Evaluating Scenario 2 (Product = 30)

If $PC \cdot PD = 30$ (from Configuration 2: P--D--C):

• Matching with Configuration A (P--A--B): $(x - 7)x = 30$. This gives the quadratic equation $x^2 - 7x - 30 = 0$. Factoring yields $(x - 10)(x + 3) = 0$. The positive solution is $x = 10$. This satisfies the condition $x > 7$. Thus, $PB = 10$ cm is a possible answer.
• Matching with Configuration B (P--B--A): $(x + 7)x = 30$. This gives the quadratic equation $x^2 + 7x - 30 = 0$. Factoring yields $(x + 10)(x - 3) = 0$. The positive solution is $x = 3$. This satisfies the condition $x > 0$. Thus, $PB = 3$ cm is a possible answer.

Final Determination of PB

Both $PB = 10$ cm and $PB = 3$ cm are mathematically valid solutions depending on the relative order of points A and B on the secant PAB.

• If $PB = 10$ cm (Configuration P--A--B), then $PA = 3$ cm. $PA \cdot PB = 3 \times 10 = 30$.
• If $PB = 3$ cm (Configuration P--B--A), then $PA = 10$ cm. $PA \cdot PB = 10 \times 3 = 30$.

Both scenarios yield a product of 30, which matches the product $PC \cdot PD = 30$ derived from the P--D--C configuration for secant PCD.

Given the options, and typically how such problems are set, the intended configuration leading to one of the multiple-choice answers is selected. The value $PB = 3$ cm corresponds to one of the options.

Therefore, the length of PB is 3 cm.