Two tangents, one length
Two ropes run from one peg T to a round stage. They touch it at P and Q. Are the ropes equal?
- From T, two tangents touch at P and Q.
- Join O to P, to Q and to T. The radii meet each tangent at 90°.
- OT is shared, and OP = OQ. So TP equals TQ.
The rule
Two tangents from one outside point are equal. TP = TQ.
Here T is the outside point and O is the centre. P and Q are the points of contact. TP and TQ are the tangent lengths.
The proof
Join OP, OQ and OT. The radii meet the tangents at right angles. So triangle OPT has a right angle at P. Triangle OQT has one at Q.
In each triangle, OT is the hypotenuse. Pythagoras gives two lines:
TP² = OT² − OP² TQ² = OT² − OQ²
OP and OQ are radii, so OP = OQ. The right-hand sides are equal. So TP² = TQ². Then TP = TQ.
A second view
The same triangles match by RHS. That means right angle, hypotenuse, side. Each has a right angle. Each has the hypotenuse OT. The side OP = OQ. So the triangles are congruent, which means the same shape and size.
Matching parts are equal. So TP = TQ. Also angle OTP = angle OTQ. OT cuts the angle at T in half.
Drag T. Watch TP and TQ.
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