Circle Theorems Questions - MME - Maths Made Easy€¦ · 6 Points , , , and are all on the circumference of the circle and is the diameter. (Level 7) Point 𝑋is the intersection

Circle Theorems

(Level 6 - 9)

2

1 Points 𝐴𝐴, 𝐵𝐵, and 𝐶𝐶 are all on the circumference of the circle. (Level 6)

𝑂𝑂 represents the centre.

Calculate the angle 𝑥𝑥, giving a reason for your answer.

[2 marks]

Answer

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2

𝐴𝐴

𝐶𝐶

𝐵𝐵𝑥𝑥98° 𝑜𝑜

3

2 Points 𝐴𝐴, 𝐵𝐵, and 𝐶𝐶 lie on the circumference of a circle. (Level 6)

The line 𝐵𝐵𝐶𝐶 passes through the centre of the circle, 𝑂𝑂.

Calculate the angle 𝑥𝑥, giving your reasoning for each step.

[2 marks]

Answer

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2

𝐴𝐴

𝐶𝐶𝑥𝑥32°

𝑜𝑜 𝐵𝐵

4

3 The diagram below shows a cyclic quadrilateral 𝐴𝐴𝐵𝐵𝐶𝐶𝐴𝐴. (Level 6)

Points 𝐴𝐴,𝐵𝐵,𝐶𝐶 and 𝐴𝐴 touch the circumference of the circle.

Line 𝐵𝐵𝐴𝐴 goes through centre 𝑂𝑂.

Work out the size of the angle marked 𝑥𝑥.

Explain your reasoning carefully.

[2 marks]

Answer

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2

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33°

𝑥𝑥

57°

𝐴𝐴

𝐵𝐵

𝐶𝐶

𝐴𝐴

𝑂𝑂

5

4 Points 𝐴𝐴, 𝐵𝐵, and 𝐶𝐶 are all on the circumference of a circle. (Level 6)

𝑂𝑂 represents the centre.

Calculate the angle 𝑥𝑥, giving your reasoning for each step.

[2 marks]

Answer

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2

𝐴𝐴

𝐵𝐵𝑥𝑥67°

𝑂𝑂

𝐶𝐶

22°

6

5 Points 𝐴𝐴, 𝐵𝐵, and 𝐶𝐶 are all on the circumference of the circle. (Level 7)

𝑂𝑂 represents the centre.

𝐴𝐴𝐴𝐴 and 𝐴𝐴𝐶𝐶 are tangents to the circle.

Angle 𝐶𝐶𝐴𝐴𝑂𝑂 = 22°

Calculate the angle 𝑥𝑥, giving your reasoning for each step.

[4 marks]

Answer

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4

𝐴𝐴

𝐶𝐶

𝑥𝑥22°

𝑜𝑜 𝐴𝐴𝐵𝐵

7

6 Points 𝐴𝐴, 𝐵𝐵, 𝐶𝐶, and 𝐴𝐴 are all on the circumference of the circle. (Level 7)

Point 𝑋𝑋 is the intersection between line 𝐴𝐴𝐶𝐶 and line 𝐴𝐴𝐵𝐵

Angle 𝐶𝐶𝑋𝑋𝐵𝐵 = 110°

Angle 𝑋𝑋𝐴𝐴𝐵𝐵 = 22°

Angle 𝐵𝐵𝐶𝐶𝑋𝑋 = 23°

6(a) Calculate the angle 𝑋𝑋𝐵𝐵𝐶𝐶

[1 mark]

Answer

6(b) Calculate the angle 𝐴𝐴𝐴𝐴𝑋𝑋

Give a reason for your answer.

[2 marks]

Answer

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3

𝐵𝐵

𝐴𝐴

𝐴𝐴

23°

22°

𝐶𝐶

𝑋𝑋110°

8

7 Points 𝐴𝐴, 𝐵𝐵, and 𝐶𝐶 are all on the circumference of the circle. (Level 8)

𝑂𝑂 represents the centre.

Angle 𝐴𝐴𝑂𝑂𝐵𝐵 = 2𝑥𝑥 + 28

Angle 𝐴𝐴𝐶𝐶𝐵𝐵 = 3𝑥𝑥 − 70

Calculate the value of 𝑥𝑥.

[3 marks]

Answer

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3

𝐴𝐴

𝐴𝐴

𝐵𝐵

2𝑥𝑥 + 28° 𝑜𝑜3𝑥𝑥 − 70°𝐶𝐶

9

8 Points 𝐴𝐴, 𝐵𝐵 and 𝐶𝐶 are on the circumference of a circle, with centre 𝑂𝑂. (Level 9)

Points 𝐶𝐶,𝐴𝐴 and 𝐸𝐸 lie on a tangent line.

𝐴𝐴𝐵𝐵 = 𝐴𝐴𝐶𝐶

Calculate angle 𝐶𝐶𝐴𝐴𝑂𝑂.

[5 marks]

Answer

End of Questions

END

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5

𝐴𝐴

𝐴𝐴

𝐶𝐶

𝐵𝐵𝑜𝑜

107° 46°

𝐸𝐸𝐹𝐹