Question

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

Explanation:

Step1: Apply the Chords - Intersecting 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: \(9x+9 = 15x-15\).
Subtract \(9x\) from both sides: \(9=15x - 9x-15\), so \(9 = 6x-15\).
Add \(15\) to both sides: \(9 + 15=6x\), i.e., \(24 = 6x\).
Divide both sides by \(6\): \(x=\frac{24}{6}=4\).

Step3: Calculate the Length of \(MN\)

Since \(MN=(x - 1)+15\), substitute \(x = 4\) into the formula. \(MN=(4 - 1)+15\).
\(MN=3 + 15=18\)

Answer:

18 units