1. Recapping: Vector addition.
ABCD is a trapezium with BA = c and BC = a. AD is twice as long as and parallel to BC . The points E and F are midpoints of AD and CD respectively.
Find the following vectors in terms of a and c:
a) 𝐴𝐶 b) CD c) EF
d) Show that CA is parallel to and twice as long as EF .

2. Vectors and Geometry
Learning Objective: Prove geometrical properties about shapes using vectors.
The diagram shows a triangle ABC with 𝐴𝐵 =a and 𝐴𝐶 =𝑏. D is a point on BC such that BD : DC = 3 : 1.
Express the following vectors in terms of a and b.
a) 𝐵𝐶 b) 𝐵𝐷
c) 𝐷𝐶 d) 𝐴𝐷
XYZ is a triangle with 𝑋𝑌 =𝑎 and 𝑋𝑍 =𝑏. Q is a point on 𝑌𝑍 such that YQ : QZ = 1 : 2 and P is a point on XZ such that XP : PZ = 2 : 1.
i) Sketch the triangle given the above information.
ii) Express the following vectors in terms of a and b.
a) 𝑌𝑍 b) 𝑌𝑄 c) 𝑃𝑄

3. Vectors and Geometry
Learning Objective: Prove geometrical properties about shapes using vectors.
ABCD is a quadrilateral. P, Q, R and S are midpoints of the lines AC, DB, AD and CB respectively. AD = a, AB = b and AC = c.
a) Find and simplify expressions in terms of a, b and c, for:
i) 𝑅𝑃 ii) 𝐴𝑄 iii) 𝑂𝑆
b) Use your answers to part a to find and simplify and expression for 𝑄𝑆.
c) What type of quadrilateral is PQRS?

4. Vectors and Geometry
Learning Objective: Prove geometrical properties about shapes using vectors.
PQRS is a quadrilateral. W, X, Y and Z ae the mid-points of PQ, QR, RS and PS respectively. 𝑃𝑄 =𝑎, 𝑄𝑅 =𝑏, 𝑅𝑆 =𝑐 and 𝑆𝑃 =𝑑.
a) Show that WX is parallel to PR
b) Show that ZY is parallel to PR
c) Show that XY is parallel to WZ
d) What can you deduce about quadrilateral WXYZ?