Tangents and Intersecting Chords
334. The Twin Tangents · equal lengths, equal angles, one line of symmetry
PA equals PB, and angles at O and P are split equally by OP.
The two tangents from an external point to a circle are equal in length (PA = PB). The line to the centre, OP, is an axis of symmetry that bisects both the angle between the tangents and the angle at the centre.
What this lesson covers
Try to break it
Drag P closer to or farther from the circle. The two tangents from P to the circle always come out equal in length (PA = PB), and OP bisects ∠APB. Try to find a P where one tangent is longer than the other; impossible. The two right triangles OAP and OBP are congruent by RHS (OA = OB = r, common OP), so PA = PB.
How you build it
Draw two tangents from an external point P.
- Draw a circle with centre O using the circle tool.
- Mark a point P outside the circle using the point tool.
- Draw radii OA and OB to the points where tangents will touch.
- Draw tangents PA and PB from P to the circle.
- Join O and P to complete the figure.
The proof, step by step
Prove that the two tangents drawn from an external point are equal in length.
- OA = OB (Radii of the same circle)
- ∠OAP = ∠OBP = 90° (Angle between radius and tangent is 90°)
- OP = OP (Common side)
- ΔAOP ≅ ΔBOP (by RHS congruence rule)
- PA = PB, ∠AOP = ∠BOP, ∠APO = ∠BPO (CPCT)
Worked example
From a point P, 13 cm away from the centre O of a circle of radius 5 cm, a tangent PQ is drawn. Find the length of PQ.
In right ΔOQP, OQ² + PQ² = OP². 5² + PQ² = 13² → PQ² = 169 − 25 = 144 → PQ = 12 cm.
- 8 cm
- 10 cm
- 12 cm — correct
- 14 cm