Sample Copyright. Academic Group. 15 Geometric Proofs using Vectors. Calculator Assumed. 1. [6 marks: 2, 2, 2,]
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1 alculator ssumed 1. [6 marks: 2, 2, 2,] Given that a and b are non-parallel vectors. Find α and β if: (a) 2a + (β 2) b = (1 α) a (b) α (3a 4b) = 6a + βb (c) αa + 5b is parallel to 3a + βb 2. [4 marks: 1, 1, 2] is a parallelogram. = a and = c. L, M, N and P are the midpoints of,, and respectively. (a) Find LM in terms of a and c. (b) Find PN in terms of a and c. cademic Group (c) Hence, use a vector method to show that LMNP is a parallelogram. c L a P M N.T. Lee 101
2 alculator ssumed 3. [8 marks: 2, 4, 2] is a trapezium with = 3. = a and = c. E and F are midpoints of and respectively. (a) Find E in terms of a and c. (b) Find EF in terms of a and c. (c) Prove that EF is a parallelogram. cademic Group E F.T. Lee 102
3 alculator ssumed 4. [14 marks: 2, 2, 5, 3, 2] is a triangle with = a and = b. D, E and F are the midpoints of,, and respectively. M = αd and MF = βf. (a) Find D and F in terms of a and b. (b) Find M and MF in terms of a and b. (c) Use your answers in (b) to find α and β. (d) Show that M = µe giving the value of µ. cademic Group (e) omment on the significance of the location of M in terms of the lines E, D and F. F E M D.T. Lee 103
4 alculator ssumed 5. [8 marks: 2, 4, 2] In triangle, K divides in the ratio λ:µ (that is K:K = λ:µ ). (a) Find K in terms of and. (b) Hence, or otherwise. prove that K = 1 [λ + µ]. λ + µ (c) Use the result above to find the position vector of a point that divides the line connecting (1, 2) to (6, 12) in the ratio 2: 3. cademic Group K.T. Lee 104
5 alculator ssumed 6. [8 marks] is a parallelogram. The point E divides in the ratio α : β. That is, the point E is such that β E =. E extended meets α+β the extended at F. Use vector methods to prove that: rea of FE = β rea of F. α+β [Hint: Let EF = λf and F = µf. ] 2 cademic Group c a E F.T. Lee 105
6 alculator ssumed 7. [4 marks] is a rhombus. = a and = c. Use a vector method to show that the diagonals of a rhombus are perpendicular to each other. 8. [8 marks: 1, 3, 4] is a right angled triangle with = 90 o. = a and = b. M is the midpoint of. (a) Explain why a. b = 0. (b) Find M 2 in terms of a and b, where a = a and b = b. (c) Hence, prove that M is the centre of a circle passing through, and. cademic Group c b a a M.T. Lee 106
7 alculator ssumed 9. [12 marks: 2, 3, 4, 3] is an isosceles triangle with =. lso, = a and = b. (a) Show that a + b = a (b) Show that b 2 = 2 a. b. (c) Show that cos = a. b a b. cademic Group (d) Hence, using a vector method, prove that the base angles of an isosceles triangle are equal..t. Lee 107
8 alculator ssumed 10. [10 marks: 1, 2, 2, 3, 2] DM and EM are respectively the perpendicular bisectors of sides and of triangle. F is midpoint of. lso, = b, = c and MD = d. (a) Find ME in terms of b, c and d. (b) Use your answer in (a) to show that [d + 21 (c b)]. c = 0 (c) Find MF in terms of b, c and d. (d) Show that MF. = 0. cademic Group (e) State the significance of the result MF. = 0. D M F E.T. Lee 108
9 alculator ssumed 11. [5 marks: 2, 3] [TIS] is a parallelogram with = a and = c. The point K divides in the ratio 2 : 1. K extended meets the line extended at D. K = αd and D = β. D (b) Find K and K in terms of a and c. K (c) Use vector methods to prove that divides the line D in the ratio 2 : 1. cademic Group.T. Lee 109
10 alculator ssumed 12. [8 marks: 3, 5] [TIS] In PQR drawn below, the point T is the midpoint of QR. Let PT = a and TR = b. (a) Find PR and PQ in terms of a and b. (b) If T is equidistant to P and R, use a vector method to prove that QPR = 90 o. cademic Group.T. Lee 110
(b) the equation of the perpendicular bisector of AB. [3]
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