Circle theorems

A, B, and C are points on a circle, centre O.

Explain why OABC is not a cyclic quadrilateral.

Work out the value of x.

A, B, C and D are points on the circumference of a circle, centre O. BD is a diameter of the circle. EBF is a tangent to the circle.

Give a reason why angle BAD = 90°.

Write down one other angle that is 90°. Give a reason for your answer.

Write down the value of angle CAD.

Write down the value of angle EBA.

P, Q, R and S are points on the circumference of a circle, centre O. Angle SQR = 14 ° and angle SPQ = 62 °.

Find the size of angle ROQ.

(a) Angle ROQ =........ °

D, E and F are points on the circumference of a circle.

Line GH is a tangent to the circle at F. Line DE is parallel to line GH.

Complete these statements to prove that triangle DEF is isosceles. Give reasons for your statements. You may not need all of the lines.

Angle........ = Angle........ because........

Angle........ = Angle........ because........

Angle........ = Angle........ because........

Angle........ = Angle........ because........

Angle........ = Angle........ because........

Triangle DEF is isosceles because........

The diagram shows a circle, centre O and radius 52 cm.

AB is a chord of the circle. The line through T is a tangent to the circle.

The chord is parallel to the tangent. The perpendicular distance between the chord and the tangent is 72 cm.

Calculate the area of the shaded segment. You must show your working.

The diagram shows a circle with centre (0, 0) and a tangent at (12, 5^-). The tangent at (12, 5^-) crosses the x-axis at (p, 0).

Find the exact value of p. You must show your working.

P, Q, T and V are points on the circumference of a circle. TV is a diameter of the circle. M is a point on PQ such that PM = MQ.

Complete these sentences to show that triangle TMP is congruent to triangle TMQ.

Side PM = side MQ because it is given to you.

Angle PMT = angle........ because........

Triangle TMP is congruent to triangle TMQ because........

The diagram shows a circle, centre the origin, with the tangent to the circle at the point (4, 2).

Write down the equation of the circle.

Show that the tangent to the circle at the point (4, 2) has gradient - 2.

Find the equation of the tangent to the circle at the point (4, 2).

In this question all angles are given in degrees.

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

Find the size of angle BCD.

Find the size of angle DAB. Give a reason for your answer.

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

Points C and D lie on a tangent to the circle at C. Angle BCD = 45 °. Angle ACB = 58°. AB = 8cm.

Find the length of BC.

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

Angle PRQ = 59° and angle PQR = 67°. Line SPT is a tangent to the circle.

Work out angle a. Give a reason for your answer.

Angle a =........ ° because........

Work out angle b. Give a reason for your answer.

Angle b =........ ° because........

E, F, G and H are points on the circumference of a circle, centre O. Acute angle EHG = x °.

Complete the following, giving the values of the angles in terms of x.

Obtuse angle EOG =........ ° because........

Write down what your working in part (b)(i) has proved.

Points A, B, C and D lie on the circumference of a circle. Line AC intersects line BD at point E.

Prove that triangle AED is similar to triangle BEC.

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

Angle ACB = 90°, AC = 6.4 m and BC = 3.7 m.

Work out the circumference of the circle. You must show your working.

Points A, B and C lie on the circumference of a circle. EAF is a tangent to the circle.

Write down the value of angle BCA giving a reason for your answer.

Angle BCA = ……………….° because........

Points G, H and J lie on the circumference of a circle, centre O.

Angle GOJ = 52° and angle GJH = x °. Lines JO and GH are parallel.

Find the value of x. You must show your working.

ABC is an isosceles triangle. The sides of the triangle ABC are all tangents to a circle of radius 6 cm, centre O.

Angle BAC = 70° and BA = BC.

Show that length BO is 17.54 cm, correct to 2 decimal places.

Find the area of triangle ABC. You must show your working.

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

EF is the tangent to the circle at D.

Angle BDF = 70°.

The ratio angle BCD: angle CBD is 5: 2.

Work out angle CBD. You must show your working.

The diagram shows a circle, centre O.

Points A, B and C lie on the circumference of the circle. Line AOB is a diameter. Line DAE is a tangent to the circle. Angle CAE = 32°.

Give a reason why angle ACB is a right angle.

The radius of the circle is 8 cm.

Calculate length BC.

In the diagram,

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

• O is the centre of the circle

• angle OAB = 15°

• angle BCO = 10°.

Calculate the acute angle AOC.

In the diagram,

• E, F and D are points on the circumference of the circle

• G, D and H lie on a tangent to the circle

• angle EFD = 55°

• angle FDE = 43°.

Explain why angle HDF is 82°.

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

Angle BAD = 112° and angle DCO = 33°.

Show that angle y = 35°.

Give reasons for each stage of your working.

Work out angle z. Give reasons for your answer.

Angle z =........ ° because........

The point (- 5, 2) lies on the circumference of a circle, centre (0, 0).

Find the equation of the circle.

Work out the gradient of the tangent to the circle at (- 5, 2).

The diagram shows a circle, centre O.

Points A, B, C and D lie on the circumference of the circle. EDF is a tangent to the circle. Angle ABC = 82 ° and angle ODC = 57 °.

Work out the value of x.

Work out the value of y.

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

Angle CAD = 28° and CD = 6.4 cm. BD is a diameter of the circle.

Calculate the area of the circle.

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

PQ is a tangent to the circle at D. Angle BDQ = 72° and angle ABD = 63°.

Work out angle x. Give a reason for your answer.

Angle x =........ ° because........

Work out angle y. Give a reason for your answer.

Angle y =........ ° because........

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

Angle BAD = 52°.

Work out angle x. Give a reason for your answer.

x =........ ° reason........

Work out angle y. Give a reason for your answer.

y =........ ° reason........

A and B are points on the circumference of a circle, centre O. CA and CB are tangents to the circle.

Prove that triangle OAC is congruent to triangle OBC.