Q1.

𝐴, 𝐡, 𝐢 and 𝐷 are points on the circumference of a circle.

!(Image27)

(a) Find the size of angle ACD

(b) Give a reason for your answer

Q2.

𝐴, 𝐡, 𝐢 and 𝐷 are points on the circumference of a circle.

(a) Find the size of angle π‘₯ giving reasons for your answer.

!(Image29)

Q3.

In the diagram, 𝐴, 𝐡, 𝐢 are points on the circumference of a circle and 𝑂 is the centre of the circle.

Find the size of angle 𝐴𝐡𝐢 giving reasons for your answer.

Q4.

In the diagram, 𝐴, 𝐡, 𝐢 are points on the circumference of a circle and the line 𝐴𝐡 is the diameter.

(b) Find the size of angle π‘₯ giving reasons for your answer.

Q5.

𝐴, 𝐡, 𝐢, 𝐷 and 𝐸 are points on the circumference of a circle.

(a) Find the size of angle π‘₯ giving reasons for your answer.

(b) Find the size of angle 𝑦 giving reasons for your answer.

𝐴, 𝐡, 𝐢 and 𝐷 are points on the circumference of a circle and 𝑂 is the centre of the circle.

Q7.

𝐡 is a point on the circumference of the circle, 𝑂 is the centre of the circle and the line 𝐴𝐡 is a tangent to the circle.

(a) Find the size of angle π‘₯

(b) Find the size of angle 𝑦

Q8.

𝐴, 𝐡, 𝐢 and 𝐷 are points on the circumference of a circle and 𝑂 is the centre of the circle. The line EF is a tangent to the circle.

Given that the angle 𝐡𝐴𝐷 has a value of 2π‘₯, find the value of π‘₯.

𝐴, 𝐡, 𝐢 and 𝐷 are points on the circumference of a circle. The line 𝐸𝐹 is a tangent to the circle.

(a) Find the size of angle π‘₯

(b) Find the size of angle 𝑦

Q10.

!(Image59)

𝐴, 𝐡 π‘Žπ‘›π‘‘ 𝐢 are points on the circumference of a circle and 𝑂 is the centre of the circle. The lines 𝐴𝐡 and 𝐴𝐢 are tangents to the circle is a tangent to the circle.

Calculate the length AO, giving your answer to 1 decimal place.

Q11.

The lines 𝐴𝐡, 𝐡𝐢 and 𝐴𝐢 are tangents to the circle and 𝑂 is the centre of the circle.

Given that the area of the triangle is 452.54mΒ², find the radius of the circle to 1 decimal place.

Answers

Q1. π‘₯ = 65Β° βˆ’ 𝐴𝑛𝑔𝑙𝑒𝑠 𝑖𝑛 π‘ π‘Žπ‘šπ‘’ π‘ π‘’π‘”π‘šπ‘’π‘›π‘‘ π‘Žπ‘Ÿπ‘’ π‘’π‘žπ‘’π‘Žπ‘™
Q2. π‘₯ = 98Β° and 𝑦 = 19Β° π‘…π‘’π‘Žπ‘ π‘œπ‘›π‘  𝑖𝑛𝑐𝑙𝑒𝑑𝑒 π‘œπ‘π‘π‘œπ‘ π‘–π‘‘π‘’ π‘Žπ‘›π‘”π‘™π‘’π‘  𝑖𝑛 𝑐𝑦𝑐𝑙𝑖𝑐 π‘žπ‘’π‘Žπ‘‘π‘Ÿπ‘–π‘™π‘Žπ‘‘π‘’π‘Ÿπ‘Žπ‘™ π‘šπ‘Žπ‘˜π‘’ 180Β° π‘œπ‘Ÿ π‘Žπ‘›π‘”π‘™π‘’π‘  𝑖𝑛 π‘žπ‘’π‘Žπ‘‘ = 360Β°
Q3. π‘₯ = 22Β° βˆ’ 𝐴𝑛𝑔𝑙𝑒 π‘Žπ‘‘ π‘π‘’π‘›π‘‘π‘Ÿπ‘’ 𝑖𝑠 𝑑𝑀𝑖𝑐𝑒 π‘Žπ‘›π‘”π‘™π‘’ π‘Žπ‘‘ π‘π‘–π‘Ÿπ‘π‘’π‘šπ‘“π‘’π‘Ÿπ‘’π‘›π‘π‘’
Q4. π‘₯ = 90Β° βˆ’ 𝐴𝑛𝑔𝑙𝑒𝑠 𝑖𝑛 π‘ π‘’π‘šπ‘–π‘π‘–π‘Ÿπ‘π‘™π‘’ 𝑖𝑠 90Β° 𝑦 = 44Β° βˆ’ 𝐴𝑛𝑔𝑙𝑒𝑠 𝑖𝑛 π‘‘π‘Ÿπ‘–π‘Žπ‘›π‘”π‘™π‘’
Q5. π‘₯ = 148Β° βˆ’ π‘‚π‘π‘π‘œπ‘ π‘–π‘‘π‘’ π‘Žπ‘›π‘”π‘™π‘’π‘  𝑖𝑛 𝑐𝑦𝑐𝑙𝑖𝑐 π‘žπ‘’π‘Žπ‘‘π‘Ÿπ‘–π‘™π‘Žπ‘‘π‘’π‘Ÿπ‘Žπ‘™ π‘šπ‘Žπ‘˜π‘’ 180 π‘‘π‘’π‘”π‘Ÿπ‘’π‘’π‘ . (opposite y) 𝑦 = 32Β° βˆ’ 𝐴𝐸𝐢 = 32 π‘‘π‘’π‘”π‘Ÿπ‘’π‘’π‘  (π‘Žπ‘›π‘”π‘™π‘’π‘  𝑖𝑛 π‘‘π‘Ÿπ‘–π‘Žπ‘›π‘”π‘™π‘’)
Q6. π‘₯ = 224Β° 𝐷𝐢𝐡 = 68 π‘‘π‘’π‘”π‘Ÿπ‘’π‘’π‘ , π‘ π‘‘π‘Ÿπ‘Žπ‘–π‘”β„Žπ‘‘ 𝑙𝑖𝑛𝑒 = 180 π‘‘π‘’π‘”π‘Ÿπ‘’π‘’π‘  𝐡𝐴𝐷 = 112 π‘‘π‘’π‘”π‘Ÿπ‘’π‘’π‘ , π‘œπ‘π‘π‘œπ‘ π‘–π‘‘π‘’ π‘Žπ‘›π‘”π‘™π‘’π‘  𝑖𝑛 𝑐𝑦𝑐𝑙𝑖𝑐 π‘žπ‘’π‘Žπ‘‘π‘Ÿπ‘–π‘™π‘Žπ‘‘π‘’π‘Ÿπ‘Žπ‘™ π‘šπ‘Žπ‘˜π‘’ 180 π‘‘π‘’π‘”π‘Ÿπ‘’π‘’π‘ 
Q7. π‘₯ = 90Β°, 𝑦 = 26Β° (angle in semicircle is 90 degrees then angles in a triangle)
Q8. π‘₯ = 52
Q9. π‘₯ = 107Β° 𝑦 = 42Β°
Q10. 25.5π‘π‘š (π‘šπ‘Žπ‘˜π‘’ π‘Ž π‘˜π‘–π‘‘π‘’ π‘Žπ‘›π‘‘ 𝑒𝑠𝑒 π‘‘π‘Ÿπ‘–π‘”π‘œπ‘›π‘œπ‘šπ‘’π‘‘π‘Ÿπ‘¦ π‘‘π‘œ 𝑓𝑖𝑛𝑑 π‘‘β„Žπ‘’ β„Žπ‘¦π‘π‘œπ‘‘π‘›π‘’π‘’π‘ π‘’)
Q11.

𝐢 = 45°