Circle theorem proofs: tangents from an external point
Circle theorems
Circle theorem proofs: tangents from an external point
6 self-marking GCSE practice questions on Circle theorem proofs: tangents from an external point, written specifically for AQA, Pearson, OCR across Higher tier. Instant marking with full worked solutions on every question.
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Question 13 marksGrade 8-9
TA and TB are two tangents drawn from an external point T to a circle with centre O, touching the circle at A
What are the common mistakes in Circle theorem proofs: tangents from an external point?
When a pupil makes one of these on Revision Workshop, the marking engine names the mistake and explains it, instead of just marking the answer wrong.
Measures rather than proves the required result
Measured the two tangent lengths in a specific diagram and found them approximately equal, rather than proving the result in general using congruent triangles
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and
B
respectively. Prove that
TA=TB
(tangents drawn from the same external point are equal in length).
Show the worked solution
Draw radii OA and OB, and the line OT from the centre to the external point.
Since TA and TB are tangents, angle OAT= angle OBT=90° (tangent perpendicular to the radius at the point of contact). M1
Consider triangles OAT and OBT: angle OAT= angle OBT=90°; OA=OB
By RHS (right angle, hypotenuse, side), triangles OAT and OBT are congruent. M2
Since the triangles are congruent, corresponding sides are equal: TA=TB.
This holds for any external point T and any circle, proving tangents from the same external point are always equal in length. B1