Apply angle and tangent theorems with reasons.
Example 1
A,B and C lie on a circle. The minor central angle AOB is 90∘. C lies on the major arc AB. Find angle ACB in degrees.
Move C along the major arc. A, B and the central angle stay fixed.Show worked answer
Answer: 45
- The circumference angle subtending the same arc AB is half the central angle.
- Angle ACB=290=45∘.
Example 2
Two tangents PA and PB touch a circle at A and B. The minor angle AOB at the centre is 60∘. Find angle APB in degrees.
A and B are the points of contact. Rotate the view to inspect the same construction.Show worked answer
Answer: 120
- Radii OA and OB are perpendicular to the tangents.
- In quadrilateral OAPB, angle P=360∘−90∘−90∘−60∘=120∘.
Example 3
A,B and C lie on a circle. The minor central angle AOB is 80∘. C lies on the major arc AB. Find angle ACB in degrees.
Move C along the major arc. A, B and the central angle stay fixed.Show worked answer
Answer: 40
- The circumference angle subtending the same arc AB is half the central angle.
- Angle ACB=280=40∘.