Properties of Circles — Study Notes

Draw any straight line across a circle and something surprising happens: the shortest path from the centre to that line always lands at the exact midpoint. That single fact turns a whole family of circle questions into ordinary right-triangle arithmetic — provided you know where to draw the triangle.

Chords and the Hidden Right Triangle

A chord is a straight segment joining two points on a circle's edge. Picture folding the circle exactly in half along a line through the centre: whatever shape sits on one side lands perfectly on the other, because a circle is identical in every direction from its middle point. A chord is just a flat cut through that shape, so the fold line — the one running from the centre straight down onto the chord at a right angle — must land exactly halfway along it. That's the key relationship: connect the centre to the chord's midpoint, and you get a line meeting the chord at 90°.

This matters because it hands you a ready-made right triangle every time: one leg is half the chord, the other leg is the unknown distance to the centre, and the hypotenuse is simply the radius (a straight line from centre to circle edge is always the radius, wherever it lands). Spot a question naming a chord's length and a circle's radius, and Pythagoras' rule is usually the fastest route to whatever's missing.

Worked Example — Distance from Centre to a Chord

1. A circle has radius 17 cm. A chord measures 30 cm. Find how far the chord sits from the centre.
2. Halve the chord first, since the centre-to-chord line always meets it at the midpoint: 30 ÷ 2 = 15 cm.
3. Build the right triangle: radius 17 cm as hypotenuse, half-chord 15 cm as one leg, unknown distance d as the other leg.
4. Apply Pythagoras: d = √(17² − 15²) = √(289 − 225) = √64 = 8 cm.

Notice the halving step happens before the square root, not after — skipping it is the single most common slip on this type of question.

Tangents bring their own right angle at the point of contact, plus a matching pair of angle rules for shapes touching the circle from inside — covered with a full audio walkthrough, the remaining worked examples, and a self-check worksheet in the full lesson below.