Determine This

Grade 11 Euclidean Geometry — How to Identify the Question Type

Grade 11 Euclidean Geometry is circle geometry — a set of theorems about chords, angles and tangents inside a circle. The hardest part isn't the calculation, it's identifying WHICH theorem applies to the diagram in front of you, and then stating the reason correctly. A calculation without the matching reason earns no marks.

Type 1: Line from the centre to a chord

Trigger words: A line is drawn from the centre, either perpendicular to a chord or to its midpoint

Trigger structure: Look for a radius or a line through the centre meeting a chord — if it's perpendicular to the chord, it bisects it, and vice versa.

Method (no numbers — just the steps)

  1. Identify the chord and the line from the centre
  2. State the reason: 'line from centre perpendicular to chord bisects the chord' (or the converse)
  3. Use the resulting equal lengths (and Pythagoras, if a radius is also involved) to calculate the unknown
Grade 11Paper 2Level 2 — Routine Procedures
O is the centre of the circle. OM ⊥ AB, OA = 10 cm, and AM = 6 cm. Calculate the length of MB, and give a reason for your answer.

Practice question — not sourced from a past paper.

Common mistake

Calculating the correct length but giving a generic reason like 'symmetry' instead of the specific circle theorem the diagram demonstrates.

See the progression — same type, increasing difficulty

Easy
O is the centre of the circle, and OM is drawn perpendicular to chord PQ, with PM = 5 cm. Calculate the length of PQ.

Practice question — not sourced from a past paper.

Medium
O is the centre of a circle with radius 13 cm. A chord AB is 24 cm long, and OM is drawn perpendicular to AB. Calculate the length of OM.

Practice question — not sourced from a past paper.

Hard
O is the centre of a circle. Two parallel chords AB and CD are 16 cm and 12 cm long respectively, and the radius is 10 cm. Both chords are on the same side of the centre. Calculate the distance between the two chords.

Practice question — not sourced from a past paper.

Type 2: Angle subtended by a diameter

Trigger words: "AB is a diameter", a triangle is inscribed with one side passing through the centre

Trigger structure: A triangle inscribed in a circle where one side of the triangle is a diameter.

Do not confuse with: Angles in the same segment (Type 3) — this theorem applies specifically when the chord in question is a DIAMETER.

Method (no numbers — just the steps)

  1. Confirm the chord is a diameter (it passes through the centre)
  2. State the reason: 'angle in a semi-circle' (the angle subtended by a diameter at the circle is 90°)
  3. Use that right angle together with other triangle properties to find the unknown
Grade 11Paper 2Level 2 — Routine Procedures
AB is a diameter of the circle with centre O. C is a point on the circle, and angle BAC = 35°. Calculate the size of angle ACB, and give a reason.

Practice question — not sourced from a past paper.

Common mistake

Applying the 90° rule to a chord that looks like it might pass through the centre without confirming this in the given information.

See the progression — same type, increasing difficulty

Easy
PQ is a diameter of a circle, and R is a point on the circle with angle PQR = 40°. Calculate the size of angle PRQ.

Practice question — not sourced from a past paper.

Medium
AB is a diameter of a circle with centre O, and C is a point on the circle such that AC = BC. Determine the size of angle ABC.

Practice question — not sourced from a past paper.

Hard
AB is a diameter of a circle with centre O. C and D are points on the circle on the same side of AB, with angle CAB = 20° and angle DBA = 35°. Calculate the size of angle between chords AD and BC where they intersect inside the circle.

Practice question — not sourced from a past paper.

Type 3: Angles in the same segment

Trigger words: Two (or more) angles are subtended by the SAME chord, from the SAME side of it

Trigger structure: Two inscribed angles standing on the same chord, with their vertices on the same arc.

Do not confuse with: The angle at the centre theorem — if one of the two angles is at the CENTRE of the circle rather than on its circumference, this is a different (though related) theorem.

Method (no numbers — just the steps)

  1. Identify the chord that both angles are subtended from
  2. Confirm both angles' vertices lie on the same side of that chord
  3. State the reason: 'angles subtended by the same chord, in the same segment, are equal'
  4. Set the two angles equal to each other
Grade 11Paper 2Level 2 — Routine Procedures
A, B, C and D lie on a circle, with C and D on the same side of chord AB. Angle ACB = 50°. Calculate the size of angle ADB, and give a reason.

Practice question — not sourced from a past paper.

Common mistake

Assuming two inscribed angles are equal without first checking they're on the SAME side of the chord — angles in opposite segments are supplementary, not equal.

See the progression — same type, increasing difficulty

Easy
P, Q, R and S lie on a circle, with R and S on the same side of chord PQ. Angle PRQ = 62°. Calculate the size of angle PSQ.

Practice question — not sourced from a past paper.

Medium
A, B, C and D lie on a circle. AC and BD intersect at E inside the circle. Angle BAC = 40° and angle ABD = 35°. Calculate the size of angle AED.

Practice question — not sourced from a past paper.

Hard
A, B, C and D lie on a circle in that order. Angle ADB = 30° and angle BDC = 45°. AB = BC. Calculate the size of angle ABD, with reasons.

Practice question — not sourced from a past paper.

Words like determine and hence appear throughout this topic — see the instruction word glossary for full definitions.