Question

ad and mn are chords that intersect at point b. what is the length of line segment mn? 4 units 6 units 18 units 24 units

Explanation:

Step1: Apply the intersecting chords theorem

When two chords intersect each other inside a circle, the products of the lengths of their segments are equal. So, \(AB\times BD=MB\times BN\). Given \(AB = 9\), \(BD=x + 1\), \(MB=x - 1\), and \(BN = 15\). Then \(9\times(x + 1)=(x - 1)\times15\).

Step2: Solve the equation for \(x\)

Expand the equation: \(9x+9 = 15x-15\).
Rearrange terms: \(15x-9x=9 + 15\).
Simplify: \(6x=24\), so \(x = 4\).

Step3: Calculate the length of \(MN\)

Since \(MN=MB+BN\), and \(MB=x - 1\), \(BN = 15\). Substitute \(x = 4\) into \(MB\): \(MB=4 - 1=3\). Then \(MN=3 + 15=18\).

Answer:

18 units