Question

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

Explanation:

Step1: Apply intersecting chords theorem

When two chords intersect, the product of the segments of one chord equals the product of the segments of the other. So:
$$BH \times HD = FH \times GH$$
Substitute values: $a \times 8 = 6 \times 6$

Step2: Solve for $a$

Calculate right-hand side: $6 \times 6 = 36$
Rearrange to solve for $a$:
$$a = \frac{36}{8} = 4.5$$

Step3: Find length of BD (diameter)

Diameter $BD = BH + HD = a + 8$
Substitute $a=4.5$:
$$BD = 4.5 + 8 = 12.5$$

Answer:

12.5 units