June 2023 Paper 3 Q6

OCR MEICurrent spec10 marksCo-ordinate GeometryVectors

6

(a) Quadrilateral KLMN has vertices K \((-4, 1)\), L \((5, -1)\), M \((6, 2)\) and N \((2, 5)\), as shown in Fig. 6.1.
Fig. 6.1: shaded quadrilateral KLMN on x- and y-axes with K(−4, 1), L(5, −1), M(6, 2), N(2, 5); P on KL (on the x-axis), Q on LM, R on MN and S on NK
Fig. 6.1
(i) Find the coordinates of the following midpoints.
  • P, the midpoint of KL
  • Q, the midpoint of LM
  • R, the midpoint of MN
  • S, the midpoint of NK
[2]
(ii) Verify that PQRS is a parallelogram. [3]
(b) TVWX is a quadrilateral as shown in Fig. 6.2.

Points A and B divide side TV into 3 equal parts. Points C and D divide side VW into 3 equal parts. Points E and F divide side WX into 3 equal parts. Points G and H divide side TX into 3 equal parts.

\(\overrightarrow{\mathrm{TA}} = \mathbf{a}\), \(\overrightarrow{\mathrm{TH}} = \mathbf{b}\), \(\overrightarrow{\mathrm{VC}} = \mathbf{c}\).

Fig. 6.2: quadrilateral TVWX with A, B on TV, C, D on VW, E, F on WX, G, H on TX; segments AH, BC, DE and GF drawn; vectors a along TA, b along TH, c along VC
Fig. 6.2
(i) Show that \(\overrightarrow{\mathrm{WX}} = k(-\mathbf{a} + \mathbf{b} - \mathbf{c})\), where \(k\) is a constant to be determined. [1]
(ii) Verify that AH is parallel to DE. [2]
(iii) Verify that BC is parallel to GF. [2]
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