Question

ac and db are chords that intersect at point h. what is the length of line segment db? 4 units 8 units 16 units 20 units

Explanation:

1. Recall the intersecting - chords theorem:

• The intersecting - chords theorem states that if two chords \(AC\) and \(DB\) intersect at a point \(H\) inside a circle, then \(AH\times HC=DH\times HB\).
• Given \(AH = 20 - x\), \(HC=x\), \(DH=x + 4\), and \(HB=12 - x\).
• So, \((20 - x)\times x=(x + 4)\times(12 - x)\).
• Expand both sides:
• The left - hand side is \(20x-x^{2}\), and the right - hand side is \(12x-x^{2}+48 - 4x=8x - x^{2}+48\).
• Set them equal: \(20x-x^{2}=8x - x^{2}+48\).
• Subtract \(-x^{2}\) from both sides of the equation, we get \(20x=8x + 48\).
• Subtract \(8x\) from both sides: \(20x-8x=48\), which simplifies to \(12x = 48\).
• Solve for \(x\): \(x = 4\).

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

• Since \(DB=DH + HB\), and \(DH=x + 4\), \(HB=12 - x\).
• Substitute \(x = 4\) into the expressions for \(DH\) and \(HB\).
• \(DH=4 + 4=8\) and \(HB=12 - 4 = 8\).
• Then \(DB=8 + 8=16\) units.

Answer:

16 units