Year 11
Use circle theorem knowledge fluently, justify geometric relationships and prove configurations using vectors.
Geometry — circles and vectors
Circle theorems describe key angle and chord properties: angle at the centre is twice the angle at the circumference (same arc); angles in the same segment are equal; the angle in a semicircle is 90°; opposite angles in a cyclic quadrilateral sum to 180°. Tangent-chord angles and alternate segment theorems provide further relationships. Vectors provide an algebraic method to prove geometric results: show equal vectors to prove parallel lines or equal lengths, use vector addition to verify collinearity, and apply dot products for perpendicularity. Combining these approaches allows rigorous proof of complex geometric configurations.
Angle at the centre vs angle at the circumference
An arc AB subtends an angle of 80° at the centre O. What angle does it subtend at point C on the circumference?
The angle at the centre is always twice the angle at the circumference subtending the same arc.
Angle in a semicircle
AB is a diameter of a circle. C is a point on the circumference. What is angle ACB?
The angle in a semicircle (subtended by a diameter) is always a right angle.
Using vectors to prove parallel lines
Given A = (1, 2), B = (3, 6), C = (2, 4), D = (4, 8). Are AB and CD parallel?
If two vectors are equal, the lines they represent are parallel and equal in length.
Cyclic quadrilateral — opposite angles sum to 180°
ABCD is a cyclic quadrilateral. Angle A = 65°. What is angle C?
In any cyclic quadrilateral, opposite angles are supplementary (sum to 180°).
P = (0, 0), Q = (3, 1), R = (5, 4), S = (2, 3). Show that PQRS is a parallelogram.
Use the examples carefully, then choose the answer.
An arc subtends an angle of 50° at the centre. What is the angle at the circumference?
💡 Angle at circumference = angle at centre ÷ 2.
ABCD is a cyclic quadrilateral. Angle B = 72°. What is angle D?
💡 Opposite angles in a cyclic quadrilateral sum to 180°.
Given A = (0, 0), B = (4, 2), C = (1, 3), D = (5, 5). Are AB and CD parallel?
💡 Vector AB = (4, 2) and vector CD = (5−1, 5−3) = (4, 2). Equal vectors mean parallel lines.
Two tangent lines are drawn from a point outside a circle to the circle. What is true about them?
💡 Two tangents from an external point to a circle are always equal in length.
P = (0, 0), Q = (3, 4), R = (6, 8). Are these points collinear?
💡 If vector QR is a scalar multiple of vector PQ, the points are collinear.
Watch for these when working through the lesson.
These ideas fit closely with this lesson.