Chords, Arcs and the Angles They Make: Quick Revision

A brief note on Chords, Arcs and the Angles They Make. Read it before a test. Try each question at the end before you open its answer.

Chords and the centre

• All radii of a circle are equal. A diameter is twice the radius, and it is the longest chord.
• Equal chords subtend equal angles at the centre, and chords that subtend equal angles at the centre are equal.
• In triangle $OAB$, $\angle OAB = \angle OBA = \dfrac{180^\circ - \angle AOB}{2}$.
• A chord equal to the radius subtends $60^\circ$ at the centre.

The perpendicular from the centre

• The perpendicular from the centre to a chord bisects the chord. The line from the centre to the middle of a chord is perpendicular to it.
• $r^2 = d^2 + \left(\dfrac{c}{2}\right)^2$ links the radius $r$, the distance $d$ and the chord $c$.
• Equal chords are equally far from the centre. A chord nearer the centre is longer.

Angles made by an arc

• The angle at the centre is double the angle at any point on the remaining part of the circle.
• Angles in the same segment are equal.
• The angle in a semicircle is $90^\circ$. The hypotenuse of a right triangle is a diameter of the circle through its corners.

A worked example

The chord $AB$ of a circle is $24\ \text{cm}$ long and the radius is $13\ \text{cm}$. Find the distance of $AB$ from the centre $O$.

1. $OM$ is perpendicular to $AB$, so $M$ is the middle point and $AM = 12\ \text{cm}$.
2. In the right triangle $OMA$, $OM^2 = 13^2 - 12^2 = 25$.
3. So $OM = 5\ \text{cm}$.

Practice with new numbers

A program to try

The program places a point $P$ at several places on a circle, with $AB$ a diameter, and measures the angle $APB$.

Mistakes to avoid

• Using the whole chord in Pythagoras. The right triangle has half the chord as a leg.
• Halving the wrong way. The angle at the centre is the big one, and the angle at the rim is half of it.
• Taking the angle in a semicircle as $180^\circ$. It is $90^\circ$.
• Thinking that a longer chord is farther from the centre. It is nearer.

Check yourself

1. A chord $AB$ subtends $50^\circ$ at the centre $O$. What is $\angle OAB$?

2. A chord $AB$ is $16\ \text{cm}$ long and lies $6\ \text{cm}$ from the centre. What is the radius?

3. An arc makes $90^\circ$ at the centre. What angle does it make at a point on the major arc?

4. The hypotenuse of a right triangle is $26\ \text{cm}$. What is the radius of the circle through its corners?