To find the angle between two tangents drawn from the same outside point, join the centre to both points of contact. The two radii meet the tangents at right angles, so the angle at the outside point and the smaller angle between the radii add to 180°.
The 2027 SEC G3 Mathematics K310 syllabus covers the angle between a tangent and a radius and the symmetry of tangents from an external point. The worked example finds both the full angle and one half of it.
- Label the centre, outside point and two points of contact.
- Mark a right angle at each point of contact.
- Use the angle sum of the four-sided figure.
- If asked for a half-angle, identify the bisector through the centre.
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Draw the radii to the points of contact
Let O be the centre and P the point outside the circle. The tangents from P touch the circle at A and B. Draw OA and OB. OA is perpendicular to PA, and OB is perpendicular to PB.
Trace O–A–P–B–O. The four interior angles add to 360°. With 90° at A and another 90° at B, the interior angles at O and P use the remaining 180°.
When a question gives a reflex angle at O, subtract it from 360° to get the smaller angle between the radii before using the quadrilateral.
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Worked example | A central angle of 124°
The smaller angle AOB is 124°. Find APB and APO. The right angles are OAP and OBP, at the places where the radius touches each tangent.
| Vertex | Interior angle | Reason |
|---|---|---|
| O | 124° | Given |
| A | 90° | Radius perpendicular to tangent |
| P | 56° | Remaining angle of the quadrilateral |
| B | 90° | Radius perpendicular to tangent |
The full angle between PA and PB is 360° − 90° − 90° − 124° = 56°. The line PO bisects APB, so APO is 28°.
To see why PO bisects the angle, compare the right triangles OAP and OBP. The radii OA and OB are equal, and OP is their common hypotenuse. The triangles are congruent, so the angles at P are equal.
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Practise from a half-angle
Suppose OPB is 33°. Since PO bisects APB, the full angle at P is 66°. Subtract 66° from 180° to get the smaller central angle of 114°.
You can check using one right triangle. In OBP the angle at O is 180° − 90° − 33° = 57°. The matching angle in the other right triangle is also 57°, so the full angle at O is 114°.
If the problem names a reflex AOB, the corresponding answer here is 360° − 114° = 246°. Write ‘smaller’ or ‘reflex’ beside your value before moving to the next part.
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Frequently asked questions
Where should I draw the right-angle mark?
At the point of contact, between the radius and the tangent.
How do I tell APB and APO apart?
Both have vertex P. APB runs from ray PA to ray PB, while APO ends at ray PO.
What if the drawing looks different from my answer?
Use the given measurements and tangent properties. The sketch shows the relationships even when the angles are drawn out of scale.
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