1.19: The points , , and are the vertices of a quadrilateral where (a) Show that is a trapezium.

4PM1/2/November/2025 — Question 9 · 16 marks

The points A\displaystyle A, B\displaystyle B, C\displaystyle C and D\displaystyle D are the vertices of a quadrilateral where
AB→=2a+4bAC→=7a+7bAD→=10a+6b\overrightarrow{AB} = 2\mathbf{a} + 4\mathbf{b} \quad \overrightarrow{AC} = 7\mathbf{a} + 7\mathbf{b} \quad \overrightarrow{AD} = 10\mathbf{a} + 6\mathbf{b}
(a) Show that ABCD\displaystyle ABCD is a trapezium.
(4)
Given that ∣a∣=1\displaystyle |\mathbf{a}| = 1 and ∣b∣=1\displaystyle |\mathbf{b}| = 1 and that a\displaystyle \mathbf{a} and b\displaystyle \mathbf{b} are perpendicular to each other
(b) find a unit vector parallel to BD→\displaystyle \overrightarrow{BD}
(4)
The point Y\displaystyle Y lies on CD\displaystyle CD such that CY:YD=1:2\displaystyle CY : YD = 1 : 2
The lines AC\displaystyle AC and BY\displaystyle BY intersect at the point X\displaystyle X
(c) (i) Use a vector method to find the ratio BX:XY\displaystyle BX : XY
(4)
(ii) Hence find, in its simplest form, the ratio
area of triangle CXY\displaystyle CXY : area of trapezium ABCD\displaystyle ABCD
(4)