Four buoys on the surface of a large, calm lake are located at A, B, C and D with position vectors given by $$\vec{OA} = \begin{pmatrix} 410 \\ 710 \end{pmatrix}, \vec{OB} = \begin{pmatrix} -210 \\ 530 \end{pmatrix}, \vec{OC} = \begin{pmatrix} -340 \\ -310 \end{pmatrix} \text{ and } \vec{OD} = \begin{pmatrix} -590 \\ -40 \end{pmatrix}$$ All values are in metres - AQA - A-Level Maths Mechanics - Question 15 - 2019 - Paper 2

To determine if quadrilateral ABCD is a trapezium, we need to find the vectors for each side:

1. Calculate Vectors AB, BC, CD, and DA:

   • ( \vec{AB} = \vec{OB} - \vec{OA} = \begin{pmatrix} -210 \ 530 \end{pmatrix} - \begin{pmatrix} 410 \ 710 \end{pmatrix} = \begin{pmatrix} -620 \ -180 \end{pmatrix} )
   • ( \vec{BC} = \vec{OC} - \vec{OB} = \begin{pmatrix} -340 \ -310 \end{pmatrix} - \begin{pmatrix} -210 \ 530 \end{pmatrix} = \begin{pmatrix} -130 \ -840 \end{pmatrix} )
   • ( \vec{CD} = \vec{OD} - \vec{OC} = \begin{pmatrix} -590 \ -40 \end{pmatrix} - \begin{pmatrix} -340 \ -310 \end{pmatrix} = \begin{pmatrix} -250 \ 270 \end{pmatrix} )
   • ( \vec{DA} = \vec{OA} - \vec{OD} = \begin{pmatrix} 410 \ 710 \end{pmatrix} - \begin{pmatrix} -590 \ -40 \end{pmatrix} = \begin{pmatrix} 1000 \ 750 \end{pmatrix} )

2. Find Gradients of AB and CD:

   • Gradient of AB: ( \frac{-180}{-620} = \frac{3}{31} )
   • Gradient of CD: ( \frac{270}{-250} = -\frac{27}{25} )

3. Check for Parallelism: Since the gradients of AB and CD are not equal, we conclude that AB and CD are not parallel.

4. Find Gradients of BC and DA:

   • Gradient of BC: ( \frac{-840}{-130} = \frac{84}{13} )
   • Gradient of DA: ( \frac{750}{1000} = \frac{3}{4} )

5. Final Conclusion:

   • Since AB is parallel to CD but not equal in length, ABCD qualifies as a trapezium.
   • As AB and CD are not equal in length, ABCD is not a parallelogram.