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 (); 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.
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: where is the centre, the point of contact and an external point.
Perpendicular from centre bisects a chord
If you drop a perpendicular from the centre 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: .
Drawn from real examiner reports.
Mixing up the named parts
A chord that happens to pass through the centre is a diameter — but a general chord does not. Students also confuse a sector (bounded by two radii and an arc) with a segment (bounded by a chord and an arc), and label an arc as the whole circumference.
Forgetting the tangent–radius right angle
Many students do not mark the between the tangent and the radius, so they cannot set up Pythagoras or SOHCAHTOA (sin = Opp/Hyp, cos = Adj/Hyp, tan = Opp/Adj). The right angle is at the point of contact, not at the external point.
Half-chord vs full chord mix-up
When the perpendicular from the centre meets a chord, the right-angled triangle uses half the chord, not the whole chord. Forgetting to halve — or forgetting to double back at the end — gives an answer that is out by a factor of 2.
Add or subtract? pick the hypotenuse first
Decide which side is the HYPOTENUSE before using Pythagoras. For a tangent length from and , is the hypotenuse, so subtract: . To find from the two shorter sides, add: .
Calculating an equal-tangent length
Two tangents from the same external point are EQUAL, so if then — just state it with the reason. Do not set up Pythagoras to "calculate" when no extra lengths are given.
Draw the radius and look for the right angle
In a tangent or chord problem, draw the radius to the contact point (tangent) or the perpendicular from the centre (chord). That gives a right-angled triangle with a clear , so Pythagoras or SOHCAHTOA (sin = Opp/Hyp, cos = Adj/Hyp, tan = Opp/Adj) finishes it.
Then choose Pythagoras or trig
Once the right-angled triangle is drawn, decide: two sides known and want the third Pythagoras; a side plus an angle involved trigonometry. Label hypotenuse, opposite and adjacent first to pick the correct ratio.
Quote the property as your reason
When a question says "give a reason", quote the property in words — e.g. "the tangent is perpendicular to the radius" or "tangents from a point are equal". The reason mark needs the named fact, not just a number.
Double back to the whole chord
With the perpendicular-from-centre fact, the triangle gives HALF the chord. If the question wants the whole chord, double your half-chord at the end; if it wants the distance from the centre, you are already there.
| Part | What it is |
|---|---|
| Centre | the middle point (often labelled ) |
| Radius | centre to the edge (plural: radii) |
| Diameter | edge to edge through the centre, |
| Chord | a straight line joining two points on the circle (not through the centre) |
| Tangent | a straight line that touches the circle at exactly one point |
| Circumference | the curved boundary all the way round |
| Arc | a part of the circumference |
| Sector | the region between two radii and an arc (a "pizza slice") |
| Segment | the region between a chord and an arc |
A diameter is just a special chord — the longest one, passing through the centre. (Arc length and sector area belong to a different spec point; here we only need to name these parts.)
What is a chord?
From a point outside a circle, two tangents are drawn touching the circle at and . The tangent has length . State the length of , and give a reason.
