Circle Theorems — Solutions

Full worked solutions, with method and final answer for every part.

Question 1

(a)The angle at the centre is twice the angle at the circumference, standing on the same arc.

70°

Question 2

(a)Opposite angles in a cyclic quadrilateral sum to 180°.

180 − 108 = 72°

Question 3

(a)This is a standard circle theorem: the tangent is always perpendicular to the radius at the point of contact.

90°

Question 4

(a)The angle subtended by a diameter at the circumference is always a right angle.

90°

Question 5

(a)By the alternate segment theorem, the tangent-chord angle equals the angle in the alternate segment.

48°

Question 6

(a)Opposite angles in a cyclic quadrilateral sum to 180°.

180 − 95 = 85°

Question 7

(a)The angle at the centre is twice the angle at the circumference on the same arc.

2 × 42 = 84°

Question 8

(a)This is a standard circle theorem.

The two tangents are always equal in length

Question 9

(a)This is a standard circle theorem.

It bisects the chord (cuts it exactly in half)

Question 10

(a)This is the 'angles in the same segment' theorem.

They are equal

Book a free trial class

Tell us a little about your requirement.