In diagram b), O is the center of the circle and AO = 2 cm. What is the measure of angle ABC and the length of AC and OB?

In diagram b), AO is the radius of the circle, so AO = 2 cm. OB is also a radius, thus OB = 2 cm. Since O is the center and AO is a diameter, the angle ABC subtends a semicircle, making it a right angle. AC is not directly determinable from the given information without more context about triangle ABC.

• Angle ABC: Since AC is a diameter, the angle subtended by the diameter at any point on the circumference is 90°. Therefore, ∠ABC = 90°.
• OB: OB is the radius of the circle. Since AO is given as 2 cm (and appears to be a radius if AC is not a diameter, but if AC is a diameter, AO is half of it. Assuming AO is a radius and AC is a chord, then OB is also a radius. If AO is a radius, OB = 2 cm. If AC is a diameter and O is the center, then AO = OC = 2 cm, and AC = 4 cm. Based on the diagram, AO appears to be a radius. Thus, OB = 2 cm.
• AC: With only the radius given, the length of chord AC cannot be determined unless it is a diameter, in which case AC = 2 * radius = 4 cm. However, the diagram doesn't explicitly state AC is a diameter. If we assume AC is a diameter and AO = 2 cm is given as a radius, then AC = 2 * AO = 4 cm. If AO = 2 cm is the radius, and AC is a chord, then AC can be any length from 0 up to 4 cm. Given that O is marked as the center and AO is a line segment with length 2cm, and OB is also a radius, we can infer OB = 2 cm. If AC is meant to be a diameter, then AC = 4 cm. However, without further clarification, AC is indeterminate. Let's assume AC is a diameter for a complete answer.

Answer: ∠ABC = 90°, AC = 4 cm, OB = 2 cm