Supplementary Worksheet 3

Supplementary Worksheet 3 Chapter 9: Vectors 1) D, E and F are the respective midpoints of the sides [BC], [CA], and [AB] of triangle ABC, where G is the center of gravity. Let P and S be two points defined by: PS = PA + PB + PC

a) Show that: SA + SB + SC = 2 SP b) i) Show that: GA + GB + GC = 0 ii) Show that: GS + 2GP = 0 . Deduce that G, S and P are collinear. iii) Construct the points P and S. 2) Consider triangle ABC. Plot the points D and E such that DA = −3DB and EC = −3EA . The parallel to (BC) through D cuts (AC) in E’. a) Show that: EC = AE =

3 AC 4

b) Deduce that [EE’] and [AC] have the same midpoint. c) Write DE ' in terms of BC d) Show that I, J and K, the respective midpoints of [AB], [DE], and [AC] are collinear. 3) ABC is a triangle. Let O be any point in the plane. Let G be the point defined by: OG =

1 (−OA + 2OB + 2OC ) . 4

a) show that : − GA + 2GB + 3GC = 0 b) M and M’ are two points in the plane defined by: MM ' = −MA + 2 MB +3MC . Show that the three points G, M and M’ are collinear. 4) Let ABCD be a quadrilateral, E the center of gravity of the triangle ABC, F and G the points defined by: EF =

1 4 ED BG = BF . 4 3

a) Show that: 1. EA + EC + ED = BD 2. GA + GC + GD = BD − 3EG 3. BD = 3EG b) Show that G is the center of gravity of triangle ACD.