Circle theorems

In the diagram below, ABCD and ABCE are cyclic quadrilaterals.

Find the size of angle x.

Calculate the size of angle y.

State the circle theorem that you have used to find your answer.

Circle Theorem:_........

In the diagram below, ABCD and ABCE are cyclic quadrilaterals.

Find the size of angle x.

Calculate the size of angle y.

State the circle theorem that you have used to find your answer.

Circle Theorem:_........

The points A, B and C lie on the circumference of a circle.

The straight line DCE is a tangent to the circle. AB = BC. ACB = x, where x is measured in degrees.

Find an expression for the size of ACD. Give your answer in terms of x.

You must give a reason for each step of your solution.

A, B, C and D are points on the circumference of a circle with centre O.

Calculate the value of x.

Circle your answer.

55° 70° 110° 125° 135°

FD is the tangent to the circle at point E, as shown below.

ABCD is a straight line. ABE = 130° and BEC = 60°.

Calculate the size of CDE.

Points A, B, C, D lie on a circle.

The line F A E is a tangent to the circle at A. B C D = 114°. D B A = x°. B A E = 2x°.

Calculate the value of x.

You must state any circle theorems that you use.

The points B, C, D and E lie on the circumference of a circle, with centre O. AF is a tangent to the circle. AO is a straight line.

Explain how you know that AOB is a right-angled triangle.

Calculate the size of angle x.

You must show all your working.

The points B, C, D and E lie on the circumference of a circle, with centre O. AF is a tangent to the circle. AO is a straight line.

Calculate the size of angle x.

You must show all your working.

What percentage of the whole circle is shaded?

You must show how you calculated your answer.

What tangent property of circles did you use in order to answer part (a)?

What percentage of the whole circle is shaded?

You must show how you calculated your answer.

What tangent property of circles did you use in order to answer part (a)?

P, Q and R are points on the circumference of a circle with centre O.

Calculate the value of x. You must state all the angle properties that you use.

You must show all your working.

P, Q and R are points on the circumference of a circle with centre O.

Calculate the value of x. You must state all the angle properties that you use.

You must show all your working.

DE is the tangent to the circle at point A, as shown below.

BC = 7 cm and AC = 13 cm. BAD = 68° and CAE = 80°.

Calculate the area of the triangle ABC.

A circle, centre O, has a radius of 4 cm. A and B are points on the circumference of the circle. Lines PA and PB are both tangents to the circle. PB = 12 cm.

What is the length of PA?

State the circle theorem you have used to find your answer.

Circle theorem:_{……........ ……..}

What is the size of PAO ?

State the circle theorem you have used to find your answer.

Circle theorem:_{……........ ……..}

Calculate the area of the quadrilateral PAOB.

The diagram below shows a circle with centre at point O.

A, B, C and D are all points on the circumference of the circle. AB = 7·5 cm and BC = 4·7 cm.

Calculate the size of angle x.

Write down the size of angle y.

State the circle theorem you have used to find your answer.

Circle theorem used:_........

A circle, centre O, has a radius of 4 cm. A and B are points on the circumference of the circle. Lines PA and PB are both tangents to the circle. PB = 12 cm.

What is the length of PA?

State the circle theorem you have used to find your answer.

Circle theorem:_{……........ ……..}

What is the size of PAO ?

State the circle theorem you have used to find your answer.

Circle theorem:_{……........ ……..}

Calculate the area of the quadrilateral PAOB.

The diagram below shows a circle with centre at point O.

A, B, C and D are all points on the circumference of the circle. AB = 7·5 cm and BC = 4·7 cm.

Calculate the size of angle x.

Write down the size of angle y.

State the circle theorem you have used to find your answer.

Circle theorem used:_........

The points A, B, C and D lie on the circumference of a circle. The straight line EDF is a tangent to the circle at D. BAC = x and CDE = 2x, where x is measured in degrees.

Find the size of BCD in terms of x. Express your answer in its simplest form.

You must give a reason for each step of your solution.

A, B and C are points on the circumference of a circle with centre O.

ACB = 74°.

Calculate the value of x. You must state any angle property of a circle that you use.

You must show all your working.

Calculate the size of angle x in the diagram below.

Calculate the size of angle x in the diagram below.

BC is the tangent to the circle at point E, as shown below.

EC = 8 cm, AC = 11 cm and DCE = 31°.

Calculate the length of AE.

Calculate the size of CED.

The line GH is a tangent to the circle at point Y.

The line EF is parallel to the line GH. The vertices of triangle EFY lie on the circle. EYG = 60°.

Prove that EFY is an equilateral triangle.

Give a reason for each step to justify your proof.

Points A, B and C lie on the circumference of a circle, centre O.

ACB = 37°.

Calculate the size of the reflex angle AOB.

A, B and C are points on the circumference of a circle.

XY is a tangent to the circle at the point A.

BAY = 74° and ABC = 53°. Prove that triangle ABC is an isosceles triangle. You must give a reason for any statement that you make or any calculation that you carry out.