Geometry & measures · Circles, arcs & sectors
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Tangents, chords & the alternate segment
Why a tangent meets a radius at 90°, why two tangents from a point are equal, why the perpendicular from the centre bisects a chord, and the alternate segment theorem.
Geometry & measures · Circles, arcs & sectors
Tangents, chords & the alternate segment
Why a tangent meets a radius at 90°, why two tangents from a point are equal, why the perpendicular from the centre bisects a chord, and the alternate segment theorem.
Why it works
A tangent is perpendicular to the radius at the point of contact. A tangent touches the circle exactly once, so every other point on it lies outside the circle — further from the centre. The point of contact is therefore the closest point of the tangent to the centre, and the shortest distance from a point to a line is always the perpendicular. So the radius meets the tangent at .Two tangents from the same external point are equal. From a point outside the circle, draw both tangents, touching at and , and join . Triangles and each have a right angle (radius–tangent), share the hypotenuse , and have equal sides (radii) — so they are congruent by RHS. Hence , and bisects both the angle and the angle . That kite shape, symmetric about , is worth recognising on sight.
The perpendicular from the centre bisects a chord. Drop a perpendicular from to a chord , meeting it at . Triangles and have a right angle, the common side , and (radii) — congruent by RHS again, so . This is the tool for chord-length questions: the radius, half the chord and the distance from the centre make a right-angled triangle, so Pythagoras finishes the job.
The alternate segment theorem. The angle between a tangent and a chord equals the angle in the alternate segment — the angle subtended by that chord from the other side of it. It follows from the earlier theorems: the tangent–chord angle and the angle at the centre both relate to the same arc, and the radius–tangent right angle converts one into the other. In practice, spot it by looking for a tangent, a chord leaving its point of contact, and a triangle inscribed on the far side of that chord.
Reasons, in full, again:
- "the angle between a tangent and a radius is 90°"
- "tangents from an external point are equal"
- "the perpendicular from the centre to a chord bisects the chord"
- "the angle between a tangent and a chord equals the angle in the alternate