Intersecting Secant Theorem Questions with Solutions

Consider the circle and the secants \( A B \) and \( C D \) such that each cuts the circle at two points.
intersecting secant and circle theorem
The intersecting secant theorem [1] states that for the two secant \( A B \) and \( C D \) intersecting a circle, there is relationship between the lengths of the segments as follows: \[ OA \times OB = OC \times OD \]

Question With Solutions

Question 1
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 secant \( OD \) and \( OB \) to write: \( \quad OA \times OD = OC \times OB \)
Substitute the known and given quantities: \( \quad 3 \times (3+x) = 2 \times (2+10) \)
Expand and simplify: \( \quad 9 + 3 x = 24 \)
Solve for \( x \): \( \quad x = \dfrac{24-9}{3} = 5\)



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



Question 3
In the figure below, \( OB \) and \( OD \) are secant 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 \( CD = 2 r \) where \( r \) is the length of the radius.
Apply the intersecting secant theorem to \( OB \) and \( OD \) to write: \( \quad OA \times OB = OC \times OD \)
Substitute the known and unknown quantities: \( \quad 4 \times (4 + 10) = 3 \times (3+2r) \)
Expand: \( \quad 56 = 9+6r \)
Group and solve for \( r \): \( \quad r = \dfrac{47}{6} \)



Question 4
In the figure below, \( OB \) and \( OD \) are secant to the circle. Find the relationship between \( x \) and \( y \) of the form \( a x + b y = c \) such that in the figure below, \( OC = 45 , OA = 42, CD = y \) and \( AB = x \).

intersecting secant theorem question 4
Solution
The intersecting secant theorem to \( OB \) and \( OD \) to write: \( \quad OA \times OB = OC \times OD \)
Substitute the known quantities: \( \quad 42 \times (42 + x) = 45 \times (45 + y) \)
Expand: \( \quad 1764 + 42 x = 2025 + 45y \)
The above may be written as: \( \quad 42 x - 45 y = 261 \)



More References and Links

The Four Pillars of Geometry - John Stillwell - Springer; 2005th edition (Aug. 9 2005) - ISBN-10 : 0387255303
Intersecting Chords Theorem Questions with Solutions
Parts of a Circle
Geometry Tutorials, Problems and Interactive Applets
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