What is the intersecting secant theorem?

The intersecting secant theorem states that when two secants intersect outside or inside a circle, the product of the lengths of the external secant segment and the entire secant segment is equal for both secants: OA * OB = OC * OD.

How do you solve problems involving intersecting secants in a circle?

You set up an equation equating the product of the external segment and the whole secant for both secants, substitute known lengths or algebraic expressions, and solve for the unknown variable.

Intersecting Secant Theorem Questions with Solutions

Theorem Overview

Consider the circle and the secants \(AB\) and \(CD\) such that each cuts the circle at two points.

intersecting secant and circle theorem

The intersecting secant theorem [1] states that for the two secants \(AB\) and \(CD\) intersecting a circle, there is a relationship between the lengths of the segments as follows:

\[ OA \times OB = OC \times OD \]

Questions With Solutions

Question 1: Find \(x = AD\) given segment lengths

Given the lengths of segments: \(OA = 3\), \(OC = 2\), and \(CB = 10\). Find \(x = AD\) in the figure below.

intersecting secant theorem question 1

Solution:

Apply the intersecting secant theorem above to the secants \(OD\) and \(OB\) to write: \[ OA \times OD = OC \times OB \]
Substitute the known and given quantities: \[ 3 \times (3 + x) = 2 \times (2 + 10) \]
Expand and simplify: \[ 9 + 3x = 24 \]
Solve for \(x\): \[ x = \dfrac{24 - 9}{3} = 5 \]

Question 2: Quadratic equation with algebraic segment lengths

Given the lengths of segments \(OA = x\), \(OC = 3\), \(CD = 13\), and \(AB = 2x\), find \(x\).

intersecting secant theorem question 2

Solution:

Apply the intersecting secant theorem to \(OB\) and \(OD\) to write: \[ OA \times OB = OC \times OD \]
Substitute the given quantities: \[ x \times (x + 2x) = 3 \times (3 + 13) \]
Expand and group like terms: \[ 3x^2 = 48 \]
Divide both sides by \(3\) and rewrite the above equation as: \[ x^2 = 16 \]
Solve for \(x\): \[ x = \pm 4 \]
\(x\) must be positive because the lengths of segments are positive, hence the solution to the given question is: \[ x = 4 \]

Question 3: Find the radius of the circle given a diameter secant

In the figure below, \(OB\) and \(OD\) are secants to the circle such that \(OD\) passes through the center \(G\) of the circle and \(OA = 4\), \(AB = 10\), and \(OC = 3\). Find the length of the radius of the circle.

intersecting secant theorem question 3

Solution:

Since \(G\) is the center of the circle, \(CD\) is a diameter and \(CD = 2r\) where \(r\) is the length of the radius.
Apply the intersecting secant theorem to \(OB\) and \(OD\) to write: \[ OA \times OB = OC \times OD \]
Substitute the known and unknown quantities: \[ 4 \times (4 + 10) = 3 \times (3 + 2r) \]
Expand: \[ 56 = 9 + 6r \]
Group and solve for \(r\): \[ r = \dfrac{47}{6} \]

Question 4: Formulate an algebraic relationship between variables

In the figure below, \(OB\) and \(OD\) are secants to the circle. Find the relationship between \(x\) and \(y\) of the form \(ax + by = c\) such that \(OC = 45\), \(OA = 42\), \(CD = y\), and \(AB = x\).

intersecting secant theorem question 4

Solution:

Apply the intersecting secant theorem to \(OB\) and \(OD\) to write: \[ OA \times OB = OC \times OD \]
Substitute the known quantities: \[ 42 \times (42 + x) = 45 \times (45 + y) \]
Expand: \[ 1764 + 42x = 2025 + 45y \]
Rearrange terms into standard form: \[ 42x - 45y = 261 \]

Theorem Overview

Questions With Solutions

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