Circle theorems, tangent angles and regular polygon exterior angles
Free examples and worked answers. No account needed.
Practise the method.
Circle theorems, tangent angles and regular polygon exterior angles
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∘.
Put it into practice.
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