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By the end of these notes, you should be able to:
Before diving in, make sure you know these words — they will appear throughout this topic.
| Term | Plain-English Definition |
|---|---|
| Circle | A perfectly round shape where every point on the edge is the same distance from the centre |
| Centre | The middle point of a circle, equidistant from every point on the circumference |
| Circumference | The curved outer edge (boundary) of a circle |
| Radius | A straight line from the centre to any point on the circumference. All radii in the same circle are equal in length |
| Chord | A straight line that connects any two points on the circumference. It does NOT have to pass through the centre |
| Diameter | A special chord that passes exactly through the centre — it is the longest possible chord |
| Tangent | A straight line that just touches the circumference at exactly one point and then carries on. It does not cross into the circle |
| Perpendicular | Meeting at exactly 90° (a right angle) |
| Bisect | To cut something exactly in half |
| Perpendicular bisector | A line that cuts another line exactly in half AND meets it at 90° |
| Equidistant | The same distance away |
| External point | A point that is outside the circle (not on the edge, not inside) |
A chord is any straight line that starts at one point on the circumference and ends at another point on the circumference. Imagine drawing a line across a circle without going through the centre — that is a chord.
What it says in simple words: If you draw a line from the centre of a circle straight down to a chord, meeting it at a right angle (90°), that line will cut the chord exactly in half.
Or put another way: The perpendicular from the centre to a chord always hits the chord at its midpoint.
The reason to state in an exam: "The perpendicular from the centre to a chord bisects the chord."
Picture a circle with centre C. Draw a chord AB somewhere inside the circle (not through the centre). Now draw a line from C down to the chord so that it meets AB at exactly 90°. Let's call the point where they meet M.
The result: AM = MB — the chord has been cut into two equal halves.
The line CM is perpendicular to AB, and M is the midpoint of AB.
This theorem is incredibly useful because it creates a right-angled triangle. You can then use Pythagoras' Theorem (a2+b2=c2) to find missing lengths such as the radius, the perpendicular distance, or half the chord length.
The right-angled triangle has:
A chord PQ has a length of 16 cm. The perpendicular distance from the centre C to the chord is 6 cm. Find the radius of the circle.
Step 1: The perpendicular from the centre bisects the chord, so half the chord = 16÷2=8 cm.
Step 2: A right-angled triangle is formed with:
Step 3: Apply Pythagoras' Theorem: h2=62+82 h2=36+64 h2=100
h=100=10 cmThe radius is 10 cm.
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