Question

line segment bd passes through the center of circle c, bh = a, and hd = 8. what is the length of the diameter? 8 units 4.5 units 12 units 12.5 units

Explanation:

Step1: Use the perpendicular - chord bisector theorem

If a line (here \(BD\)) passes through the center of a circle (\(C\)) and is perpendicular to a chord (\(FG\)), then it bisects the chord. So \(FH = HG=6\).

Step2: Apply the intersecting chords theorem

The intersecting chords theorem states that if two chords \(BD\) and \(FG\) intersect at \(H\), then \(BH\times HD=FH\times HG\). Substituting the known values: \(a\times8 = 6\times6\). So \(8a=36\), and \(a = 4.5\).

Step3: Calculate the length of the diameter

The length of the diameter \(BD=BH + HD\). Substitute \(a = 4.5\) and \(HD = 8\) into the formula. \(BD=a + 8=4.5+8=12.5\)

Answer:

12.5 units