Explore the properties of chords, arcs, and angles in a circle, including the relationship between equal chords and their distances from the centre.
A perpendicular from the centre of a circle to a chord bisects the chord. Equal chords are equidistant from the centre.
A chord of length 16 cm is 6 cm from the centre. Find the radius.
Half chord=8 cm; radius²=8²+6²=64+36=100; radius=10 cm
Two chords are 5 cm and 7 cm from the centre. Which chord is longer?
The 5 cm chord (closer to centre) is longer, since chords closer to the centre are longer
A chord of length 30 cm is 8 cm from the centre. Find the radius.
Half chord=15; radius²=15²+8²=225+64=289; radius=17 cm
The angle subtended by an arc at the centre is twice the angle subtended at any point on the remaining circumference.
An arc subtends an angle of 80° at the centre. Find the angle it subtends at a point on the remaining circumference.
80°÷2=40°
An angle at the circumference is 35°. Find the angle subtended by the same arc at the centre.
35°×2=70°
An angle at the circumference is 48°. Find the central angle for the same arc.
48°×2=96°
1. A chord of length 24 cm is 5 cm from the centre. Find the radius.
2. Two equal chords are drawn in a circle. What can be said about their distance from the centre?
3. An arc subtends 100° at the centre. Find the angle at the circumference.
4. An angle at the circumference is 42°. Find the central angle for the same arc.
5. A chord is 8 cm long and the circle's radius is 5 cm. Find its distance from the centre.
6. A chord of length 48 cm is 7 cm from the centre. Find the radius.
7. Two equal chords are drawn in a circle. What can be said about the arcs they cut off?
8. An arc subtends 110° at the centre. Find the angle at the circumference.
9. An angle at the circumference is 39°. Find the central angle.
10. A chord is 10 cm long and the circle's radius is 13 cm. Find its distance from the centre.