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 a n 9 15 b x - 1 x + 1 m d

Explanation:

1. Recall the intersecting - chords theorem:

• The intersecting - chords theorem states that if two chords \(AD\) and \(MN\) intersect at a point \(B\) inside a circle, then \(AB\times BD=MB\times BN\).
• Given \(AB = 9\), \(BN = 15\), \(MB=x - 1\), and \(BD=x + 1\).
• Substitute the values into the intersecting - chords formula: \(9\times(x + 1)=(x - 1)\times15\).
• Expand both sides: \(9x+9 = 15x-15\).
• Move the \(x\) terms to one side and the constants to the other side: \(15 + 9=15x - 9x\).
• Combine like - terms: \(24 = 6x\).
• Solve for \(x\): \(x = 4\).

1. Find the lengths of \(MB\) and \(BN\):

• Since \(x = 4\), then \(MB=x - 1=4 - 1 = 3\) and \(BN = 15\).

1. Calculate the length of \(MN\):

• \(MN=MB + BN\).
• Substitute the values of \(MB\) and \(BN\): \(MN=3 + 15=18\) units.

Answer:

18 units