In the diagram on the right, O is the centre of the circle and A, B and C are points on the circumference.
Find the measure of the following angles. (i) ∠ACB (ii) ∠ABC
2. Find the measure of ∠A in the following diagrams. (i)
(i) (i) A (i) A (i) O O O O O A A A A A (i) (ii) (ii) (ii) (ii) (ii)
54° 54° 54° 54° 54° A
(ii) A
(iii) (iii) (iii) (iii) (iii) A
(iii) A
O O O O O 43° A
43° 43° 43° 43°
(iv) (iv) (iv) (iv) (iv) A
(iv) A
O O O O O 32° 32° 32° 32° 32°
A O O O O O A A A A
78° 78° 78° 78° 78° A A
A A
O O O O O A
A
3. Find the value of the side of the triangle marked with the letter y in each of the following diagrams (where Ois the centre of each of the circles).
(i) y y y (ii)
O O O O O y
y (iii) y
17 cm 17 cm 8 cm O O O O O y
y y
17 cm 17 cm 8 cm8 cm 8 cm y
17 cm O O yO O O y y
y y
4. Write down whether each of the following inscribed shapes is a cyclic quadrilateral, a quadrilateral, or neither. (i)
(i) (i) (i) (i) (i) (ii) (ii) (ii) (ii) (ii) (ii) (iii) (iii) (iii) (iii) (iii) (iii) (iv) (iv) (iv)(iv) (iv) (iv) (v) (v) (v) (v) (v) (v) 8 cm y
(iv) y
50 m y y
O O O O O 50 m
14 m 50 m 50 m
14 m 50 m
y 14 m y 14 m 14 m
(v) y
y y y O O O O O 365 cm 365 cm365 cm 365 cm365 cm A 30° O C B
(v)
(v)
(v)
(v) (v)
(v)
5. The area of a circle is 64π cm2
. Find the circumference of the circle, correct to one decimal place. 6. Find the diameter of a circle, where the circumference of the circle is 43·96 cm (taking π = 3·14).
7. Copy the table below and put a tick (✓) in the correct box in each row to show whether each statement is always true, sometimes true, or never true. Justify your answer in each case. Statement
Tick one box only for each statement Always true Sometimes true Never true
(i) A quadrilateral has four sides. (ii)
A circle can be drawn through the four vertices (corners) of a quadrilateral.
(iii) A triangle inscribed in a circle has a right angle. (iv) A triangle’s angles sum to 180°. (v) A diameter of a circle is a chord.