Circle properties — parts of a circle, tangents and chords

Key learnings

• •

  A tangent meets the radius at 90°

  At the point where a tangent touches the circle, the radius drawn to that point is perpendicular to the tangent. This makes a right-angled triangle, so you can use Pythagoras: $OT^2 + TP^2 = OP^2$ where $O$ is the centre, $T$ the point of contact and $P$ an external point.

• •

  Know the named parts of a circle

  The centre is the middle point; a radius joins the centre to the edge; a diameter crosses the full circle through the centre ($d = 2r$); a chord joins two points on the circle (but not through the centre); a tangent touches the circle at exactly one point; the circumference is the curved boundary. An arc is part of the circumference, a sector is the "pizza slice" between two radii, and a segment is the region cut off by a chord.

• •

  The perpendicular from the centre bisects a chord

  If you drop a perpendicular from the centre $O$ onto a chord, it cuts the chord exactly in half. So the half-chord, the perpendicular distance and the radius form a right-angled triangle: $(\tfrac{1}{2}\text{chord})^2 + d^2 = r^2$.