SM2.5 Circles and tangents
You can use the two tangent theorems, that a tangent is perpendicular to the radius at the point of contact and that the two tangents from an external point are equal, to find lengths and angles.
Topics shown in module order.
A tangent touches a circle at exactly one point, and the chapter proves just two things about it: the radius to that point meets the tangent at a right angle, and the two tangents drawn from any outside point have equal lengths. Two facts sounds small, but combined with Pythagoras and the angle sum they solve a whole family of figures, and the equal-tangents fact quietly builds the right angle that many proofs need.
Work through these
Use the radius-tangent right angle with Pythagoras to find lengths in tangent figures
The right angle at the point of contact is the door Pythagoras walks through. Most numeric questions in the chapter are this one move, and recognising it on sight is the fluency being built.
Use the equal-tangents fact to chase lengths around a figure with several tangents
When a triangle or quadrilateral wraps a circle, every vertex contributes a pair of equal tangent lengths, and labelling them with the same letter unravels the figure. The labelling habit is the technique.
Follow the proof of each theorem once, and note the single idea each proof turns on
One proof turns on the shortest distance from a point to a line, the other on congruent triangles. Knowing the turning idea is what lets you reproduce a proof under examination conditions instead of reciting it.
Find angles in figures that mix tangents with triangles and quadrilaterals
The right angle at the contact point feeds the angle sum of whatever figure surrounds it. These mixed questions are where the chapter meets the rest of geometry, and they reward writing every known angle onto the figure first.
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