In diagram c), AB and AC are tangents to the circle from point A. The angle OAC is 90 degrees, the angle OAB is 90 degrees, OB = OC, angle OAB = angle OAC, AO = 7 cm, and angle OAB = 28°. What is the measure of angle BAO and the length of AC?

In diagram c), AB and AC are tangents to the circle from point A. OB and OC are radii. OA is a line segment connecting the center O to point A. The angle OAB and OAC are given as 90 degrees, which is incorrect as they are angles between the radius and the tangent at the point of tangency. The diagram shows OB is perpendicular to AB and OC is perpendicular to AC, which is correct for tangents. OA is the angle bisector of angle BAC and also bisects angle BOC. Angle OAB is given as 28 degrees in the diagram, not 90 degrees. Angle AOB and AOC are not necessarily equal to 28 degrees. The length OA is given as 7 cm. AB and AC are tangents from A, so AB = AC.

• ∠BAO: The diagram shows that ∠OAB = 28°. Since AB and AC are tangents from A, OA bisects ∠BAC. If ∠OAB = 28°, then ∠OAC = 28° and ∠BAC = 56°. The diagram indicates that OA = 7 cm. In the right-angled triangle OAB (where ∠OBA = 90°), we can use trigonometry. However, the angle given as 28° is ∠BAO, not ∠AOB. Thus, ∠BAO = 28°.
• AC: In the right-angled triangle OBA, we have ∠OBA = 90°, ∠BAO = 28°, and OA = 7 cm. We can find AB using the sine function: \( AB = OA             \sin(   ) \). Since AB and AC are tangents from the same external point, AB = AC. Therefore, \( AC = AB = OA             \sin(28^) \). \( AC             \approx 7             0.4695 \). \( AC             \approx 3.2865 \) cm.

Answer: ∠BAO = 28°, AC ≈ 3.29 cm