Grade 10 Chords of a Circle: Short Notes

10 ශ්‍රේණිය · 26 පරිච්ඡේදය · වෘත්තයක ජ්‍යා

44%

of Grade 10 term tests include it (24 of 54)

151

marks in past papers (35 questions)

10 min

full lesson with quick checks

🎯 What you'll learn

Name a chord, its midpoint and the perpendicular from the centre
State and use: the line from the centre to the midpoint of a chord is perpendicular to it
State and use the converse: the perpendicular from the centre bisects the chord
Combine the chord theorems with Pythagoras' theorem to find lengths
Use the theorems to find angles and to write short proofs

How wide is the well?

In a Kurunegala village, a 1.6 m plank lies across the round mouth of a well, touching the rim at both ends. Its middle is 0.6 m from the centre of the well. Nobody can measure across the deep water, but one circle theorem plus Pythagoras tells you the width in a minute.

What is a chord?

A chord is a straight line segment joining two points on a circle. The plank on the well is a chord.• The longest chord passes through the centre: it is a diameter.• The midpoint of chord AB is the point M with AM = MB.• OM is the line from the centre O to that midpoint.

The key picture

O is the centre and M is the midpoint of chord AB (the amber ticks show AM = MB). The red line OM meets AB at a right angle.

Theorem: centre to midpoint

The straight line joining the centre of a circle to the midpoint of a chord is perpendicular to the chord.Why? In △OAM and △OBM: OA = OB (radii), AM = BM (M is the midpoint), OM is common. So the triangles are congruent (SSS) and ∠OMA = ∠OMB. They sit on a straight line, so each is 90°.

The converse

The perpendicular drawn from the centre of a circle to a chord bisects the chord.So if OM ⊥ AB, then AM = MB.A useful result: the perpendicular bisector of any chord passes through the centre. Fold a paper circle so a chord lies on itself and the crease goes through the centre.

🧮 Chord + Pythagoras

Triangle OMA has a right angle at M. OA is the radius (hypotenuse), OM the distance from the centre and AM half the chord. Know any two, find the third.

OA² = OM² + AM² (AM = ½ AB)

✏️ Find the radius

A chord AB of length 16 cm is 6 cm from the centre O. Find the radius.

1Draw OM ⊥ AB. The perpendicular from the centre bisects the chord, so AM = 8 cm.
2In right-angled △OMA: OA² = 6² + 8² = 36 + 64 = 100
3OA = √100 = 10 cm

✅ 10 cm

⚠️ Half the chord, not all of it

The right-angled triangle uses half the chord. Putting 16 instead of 8 into Pythagoras is the top slip in term tests. Two more:• Adding sides (6 + 8 = 14) instead of squaring.• Stopping at the half chord when the question asks for the whole chord.Mark the right angle and the half chord on your sketch first.

✏️ Back to the well

The 1.6 m plank is a chord and its middle is 0.6 m from the centre. Find the diameter of the well.

1The line from the centre to the plank's midpoint is perpendicular to it. Half the plank = 0.8 m.
2r² = 0.6² + 0.8² = 0.36 + 0.64 = 1
3r = 1 m, so the diameter = 2 × 1 = 2 m.

✅ 2 m

⚠️ Give the reason, not 'it looks like'

Examiners report students who can't state that OM ⊥ AB when M is the midpoint, or who miss that OA = OB makes △OAB isosceles. Never assume a right angle because it looks right. Write the reason in words: 'line joining centre to midpoint of chord' or 'perpendicular from centre bisects chord'.

📌 Remember

• Centre to midpoint of a chord ⇒ perpendicular to the chord.• Perpendicular from the centre ⇒ bisects the chord.• Right triangle: radius² = distance² + (half chord)².• OA = OB (radii) gives an isosceles triangle: use it for angles.• In proofs, write the theorem as your reason.

🧾 Quick revision sheet

What is a chord?	A straight line segment joining two points on a circle.
The line from the centre to the midpoint of a chord is...	...perpendicular to the chord.
The perpendicular from the centre to a chord...	...bisects the chord.
Which is the longest chord of a circle?	A diameter.
Radius r, distance d from the centre: half the chord?	√(r² − d²)
Where does the perpendicular bisector of a chord go?	Through the centre of the circle.

Now test yourself: 4 quick checks and 2 more worked examples

The full lesson checks your answers as you go, and Gedara Sir tracks which parts you've got.

📖 Start the 10-minute lesson free ▶ Practise 5 past-paper questions