The value of the variable <em>x</em> can be determined using the circle theorem to
determine the angle formed by <em>x</em>.
The correct responses are;
- (1) x = <u>30°</u>
- (2) x = <u>100°</u>
- (3) x = <u>60°</u>
Reasons:
(1) From circle theorem, we have;
- Equal chords subtends equal arcs and vice versa
- Equal chords subtend equal angles at the center of a circle
Therefore;
Angle subtended by at center = Angle subtended by
at the center;
- Angle at the center = 2 × Angle at the circumference
Therefore;
Angle subtended by at the circumference = Angle subtended by
at the circumference
Which gives;
- x = <u>30°</u>
(2) given
Angle subtended by arc at the center = x
Equal arcs subtend equal angles at the center of a circle.
Therefore;
Angle subtended by arc at the center = x
Angle at center = 2 × angle at circumference
Therefore;
x = 2 × 50° = 100°
x = <u>100°</u>
(3) given
By definition, we have;
Angle subtended by at the center = 3 × Angle subtended by
at the center
According to circle theorem, we have;
Angle at the center = 2 × Angle at the circumference
Let <em>y</em> represent the angle subtended at the circumference by , we have;
y = 20°
= 2·y
= 3 ×
= 3 × 2·y = 6·y
Therefore;
= 6 × 20° = 120°
= 120°
= 2·x
Therefore;
2·x = 120°
x = <u>60°</u>
Learn more circle theorems here: