Drop a line from the centre
The perpendicular from the centre bisects the chord
Where does it land?
Draw a chord AB. Now drop the shortest line from the centre O to the chord. It meets AB at a point F, at a right angle.
Where does F land on the chord AB?
What this lesson covers
The idea
The perpendicular from the centre of a circle to a chord bisects the chord.
Where does it land?
Draw a chord AB. Now drop the shortest line from the centre O to the chord. It meets AB at a point F, at a right angle.
Where does F land on the chord AB?
- Exactly in the middle
- Nearer one end
- It depends on the chord
Move the foot
Drag F anywhere inside the circle. The chord AB runs through F at a right angle to OF. Compare AF and FB.
F is the midpoint
The perpendicular from the centre of a circle to a chord bisects the chord.
In the right-angled triangles OFA and OFB, the hypotenuses OA and OB are equal (radii) and OF is common. So the triangles are congruent (RHS), and AF = FB.
Notes
The perpendicular from the centre of a circle to a chord bisects the chord.
Check yourself
OF is the perpendicular from the centre O to chord AB, and AB = 12 cm. What is AF, in cm?
Answer: 6 cm
F is the midpoint of AB, so AF = 12 ÷ 2 = 6 cm.
A circle has radius 5 cm. The perpendicular from the centre to a chord is 3 cm long. How long is the chord, in cm?
Answer: 8 cm
In triangle OFA, AF² = 5² − 3² = 16, so AF = 4 cm. F is the midpoint, so the chord is 2 × 4 = 8 cm.
The perpendicular from the centre O to chord AB meets it at F. Which statement is true?
Why are right-angled triangles OFA and OFB congruent?
- AF = FB — correct. Yes. The perpendicular from the centre bisects the chord.
- AF = OF. Nothing makes AF and OF equal. What is equal is AF and FB.
- OF = OA. OA is a radius. It is longer than OF, because OA is the hypotenuse of triangle OFA.
- Because AF = FB is given. That is what we want to show, so we cannot use it.
- Hypotenuses OA = OB (radii) and the side OF is common (RHS) — correct. Yes. This is the RHS test.
- Because ∠AOF = ∠BOF is given. That is not given. We start from the right angle at F and the equal radii.