Question

question 10 of 10
if the blue radius below is perpendicular to the chord \\( \overline { a c } \\), which is 6 units long,
what is the length of the segment \\( \overline { a b } \\)?
a. 3 units
b. 1.5 units
c. 12 units
d. 6 units

Explanation:

Step1: Apply the perpendicular - chord theorem

The perpendicular - chord theorem states that if a radius of a circle is perpendicular to a chord, then it bisects the chord.
Given that \(AC = 6\) units and \(OB\perp AC\).
Let \(AB=x\) and \(BC=x\) (by the perpendicular - chord theorem).
Since \(AC=AB + BC\) and \(AC = 6\) units.

Step2: Solve for \(AB\)

We know that \(AC=AB+BC\) and \(AB = BC\) (because the radius perpendicular to the chord bisects the chord).
Substituting \(AC = 6\) into \(AC=AB + BC\) and using \(AB = BC\), we get \(6=AB+AB\).
Combining like terms, \(6 = 2AB\).
Dividing both sides by 2: \(AB=\frac{6}{2}\).

Answer:

A. 3 units