Optional extension: Circle theorems, tangent angles and regular polygon exterior angles
Example 1
PA and PB are tangents to a circle with centre O. The smaller angle AOB is (x+47)∘ and angle APB is (x+71)∘. Find angle OBA, in degrees.
Two tangents to a circleDiagram not to scaleShow worked answer
Answer: 51
OA and OB are perpendicular to their tangents
angles AOB and APB sum to 180∘.
Thus (x+47)+(x+71)=180, giving x=31.
Angle AOB=78∘ and angle APB=102∘.
Since OA= OB, angle OBA=2180−78=51∘.
Example 2
A,B and C lie on a circle with centre O. The acute angle between the tangent at A and chord AB is x∘. In triangle ABC, angle ABC=(2x−33)∘ and angle BAC=57∘. Find angle OAB.
Tangent and chordDiagram not to scaleShow worked answer
Answer: 38
By the alternate-segment theorem, angle ACB=x.
The triangle angle sum gives x+(2x−33)+57=180
x=52.
The smaller central angle AOB is twice angle ACB: 104∘.
Since OA= OB, angle OAB=2180−104=38∘.
Example 3
PA and PB are tangents to a circle with centre O. The smaller angle AOB is (x+122)∘ and angle APB is (3x+18)∘. Find angle APB, in degrees.
Two tangents to a circleDiagram not to scaleShow worked answer
Answer: 48
OA and OB are perpendicular to their tangents
angles AOB and APB sum to 180∘.
Thus (x+122)+(3x+18)=180, giving x=10.
Angle AOB=132∘ and angle APB=48∘.
Therefore angle APB=48∘.
Circle and Polygon Reasoning II practice | Year 12 | Ascent Arcade