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Geometric Proof

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AQA Level 2 Further Maths practice questions, each with a hint and a full worked solution.

Have a pen and paper handy, to work out your answers.

Question 11 mark

A, B and C are points on a circle.

TA is a tangent to the circle at A.

Angle TAB=y

In a proof, a student writes

angle ACB=y

Which reason should the student give?

Hint

The angle y is between a tangent and a chord, and there are no parallel lines in the diagram.

Worked solution
  1. Angle TAB is between the tangent TA and the chord AB
  2. Angle ACB is in the segment on the other side of AB (the alternate segment)
  3. So they are equal by the alternate segment theorem
  4. 'Alternate angles' needs parallel lines, which there are not here
  5. Answer: Alternate segment theorem

Question 23 marks

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

ABE is a straight line.

DA=DC

Angle CBE=x

Work out an expression for angle DAC in terms of x.

Give your answer in its simplest form.

Type powers with ^ and fractions with /, for example x^2 or (x+1)/3

Hint

Find angle ABC first, then use the fact that ABCD is a cyclic quadrilateral.

Worked solution
  1. Angle ABC=180^\circ-x (angles on a straight line add up to 180^\circ)
  2. Angle ADC=180^\circ-(180^\circ-x)=x (opposite angles of a cyclic quadrilateral add up to 180^\circ)
  3. Triangle ADC is isosceles because DA=DC, so angle DAC= angle DCA (base angles of an isosceles triangle are equal)
  4. Angle DAC=\dfrac{180^\circ-x}{2} (angles in a triangle add up to 180^\circ)
  5. Answer: 90^\circ-\dfrac{x}{2}

Question 3Challenge5 marks

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

TA is a tangent to the circle at A.

Angle TAB=3x Angle OBA=2x+15^\circ

(a)

Work out the value of x.

3 marks

(b)

Angle OAC=20^\circ

Work out the size of angle ABC.

2 marks

Hint

Radii are equal, and a tangent is perpendicular to the radius at the point of contact.

Worked solution

Part (a)

  1. OA=OB (radii), so triangle OAB is isosceles
  2. Angle OAB= angle OBA=2x+15^\circ (base angles of an isosceles triangle are equal)
  3. Angle OAT=90^\circ (the angle between a tangent and a radius is 90^\circ)
  4. So (2x+15)+3x=90
  5. 5x=75
  6. Answer: x=15

Part (b)

  1. Angle TAB=3\times15=45^\circ and angle OAB=2\times15+15=45^\circ
  2. Angle ACB=45^\circ (alternate segment theorem)
  3. Angle CAB=20^\circ+45^\circ=65^\circ
  4. Angle ABC=180^\circ-65^\circ-45^\circ=70^\circ (angles in a triangle add up to 180^\circ)
  5. Answer: 70^\circ

More on this topic: Geometric Proof worksheet with full solutions

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