Higher – Calculations using the alternate segment theorem, tangents and chords

Part ofMathsGeometry and measure

Key points about the alternate segment theorem, tangents and chords

Bullet points represented by lightbulbs

  • Circles have different angle properties, described by . This guide covers:

    • are to a at the point of intersection

    • tangents either side of circle from a point are equal in length

    • the angle between a tangent and a is equal to the angle at the circumference in the alternate (alternate segment theorem)

  • It is important to remember other angles rules, such as angles on a straight line add up to 180°, and to recognise angles in parallel lines.

Make sure you are confident in applying Pythagoras’ theorem before working with angles made by tangents and perpendiculars to chords.

When you are confident with the theorems in this guide, have a look at the guide on how to calculate angles with circles.

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Tangent theorems

A tangent is a straight line which touches the of a circle at a single point.

A tangent makes a 90° angle or is perpendicular to the radius at that point.

The tangents from a point to a circle are equal in length.

Find out more, along with a worked example

GCSE exam-style questions

A pen and a piece of paper with question marks on it.
  1. Line AB is a tangent to the circle at P.

Find 𝑥.

A geometric diagram showing a circle with its centre labelled O. A point P lies on the circle where a tangent extends to point B outside the circle. A straight line from O passes through the circle and continues to point A outside the circle, forming an angle of 2x at A. The angle at the centre O, between the radius OP and the line to A, is labelled 7x.

  1. Line EF is a tangent to the circle at X.

Line EG is a tangent to the circle at Y.

Angle FEG = 38°

Calculate the size of angle 𝑎.

A geometric diagram showing a circle with centre labelled O. Two tangents meet outside the circle at point E, forming an angle of 38°. The tangents touch the circle at points Y and X, and extend to points G and F respectively. Inside the circle, lines OY and OX are drawn from the centre to the points of contact. Another line connects Y and Z, where Z lies on the circle near X, forming an angle labelled a at Z between YZ and ZX.

  1. Line GH is a tangent to the circle of radius 5 cm at X.

GX = 12 cm

Calculate the length of GO.

A geometric diagram showing a circle with centre labelled O. A point X lies on the circle where a tangent extends through points G and H outside the circle. The radius OX is drawn and labelled 5 cm. The segment from G to X along the tangent is labelled 12 cm.

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What is the alternate segment theorem?

Every vertex of triangle PQR touches the circumference of a circle. Tangent AB, intersects at point P on the circumference. Alternate angles APR and PQR are equal.

The alternate segment theorem states that the angle between a tangent and a chord is equal to the angle at the circumference in the alternate segment.

In the image to the right, angles RPA and PQR are in alternate segments.

Every vertex of triangle PQR touches the circumference of a circle. Tangent AB, intersects at point P on the circumference. Alternate angles APR and PQR are equal.
Circle with a smaller shaded are formed from a chord, labelled minor segment and and the rest of the circle labelled as the major segment.

A chord is a straight line joining two points on the circumference of a circle.

It cuts a circle into two segments, a minor segment and a major segment.

The image to the left shows examples of alternate segments.

Find out more about the alternate segment theorem below

GCSE exam-style questions

  1. Line AB is a tangent to the circle at P.

Angle PRQ = 65°

Angle PQR = 58°

Calculate the size of angle 𝑥.

Triangle PQR is formed from point P on the tangent (AB) and R and Q on the circumference. Angle PRQ equals 65 degrees, PQR equals 58 degrees and BPQ is labelled x

  1. Line AB is a tangent to the circle at X.

The length of line ZX is equal to the length of line ZY.

Angle BXZ = 64°

Calculate the size of angle 𝑤.

Line AB is a tangent to a circle at point X. Triangle XYZ is formed within the circle and angle BXZ is 64 degrees. Angle ZXY is unknown and labelled as w

  1. Line XY is a tangent to the circle at A.

Angle BAY = 48°

Calculate the size of angle 𝑗.

Line XY is a tangent at point A on a circle. Triangle ABC is within the circle and touches the circumference. Angle YAB is 48 degrees. A smaller triangle (ABO) has an unknown angle labelled J at point O (the centre of the circle)

  1. In the diagram, lines AD and BC are parallel.

Line PQ is a tangent to the circle at A.

Angle DAP = 47°

Angle BAQ = 31°

Calculate the size of angle 𝑦.

Tangent line PO, meets a circle at point A. ABCD forms a cyclic quadrilateral. Lines AD and BC are parallel and the diagonal through the quadrilateral (AC), does not pass through the centre of the circle. Angle PAD equals 47 degrees and angle QAB equals 31 degrees. Angle ABC is labelled y

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What are the properties of chords?

A chord is a straight line joining two points on the circumference of a circle.

When the perpendicular from the centre of a circle to a chord is drawn, the line the chord.

Find out more, along with a worked example

GCSE exam-style questions

  1. The circle centred at O has a radius, OR, which bisects line BC at X.

Angle BOR = 33°

Calculate the size of angle 𝑦.

Chord line CB, extends past the circumference to point A, along the same line. This line is bisected at point X, from the centre of the circle, to the circumference, by line OR. Angle BOX is labelled as 33 degrees and angle OBA is unknown and labelled y

  1. A circle with a radius of 5 cm has a line OX which bisects a chord, AB, of length 6 cm at X.

Calculate the length of OX.

Triangle ABO is formed between the centre of the circle and its circumference. Line OX bisects the chord line AB. Both AX and BX are equal in length. AB is 6 centimetres and OB is 5 centimetres

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Check your understanding

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Quiz – Using the alternate segment theorem, tangents and chords

Practise what you've learned about calculating using the alternate segment theorem, tangents and chords, with this quiz.

Now you've revised calculating with the alternate segment theorem, tangents and chords, why not look at area of circles and sectors?

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