Vectors, SPM Practice (Paper 2)


Question 4:
Diagram below shows quadrilateral ABCD. The straight line AC intersects the straight line BD at point E.



It is given that BE:ED=2:3,  AB =10 x ˜ ,  AD =25 y ˜  and  BC = x ˜ +15 y ˜ . (a) Express in terms of  x ˜  and  y ˜ , (i)  BD (ii)  AE (b) Find the ratio AE:EC.

Solution:
(a)(i)
BD = BA + AD   = AD AB   =25 y ˜ 10 x ˜

(a)(ii)
AE = AB + BE   = AB + 2 5 BD   =10 x ˜ + 2 5 ( 25 y ˜ 10 x ˜ )   =10 x ˜ + 2 5 ( 25 y ˜ 10 x ˜ )   =10 x ˜ +10 y ˜ 4 x ˜   =6 x ˜ +10 y ˜   =2( 3 x ˜ +5 y ˜ )

(b)
EC = EB + BC   = BC BE   = BC 2 3 ED   = BC 2 3 ( EA + AD )   = x ˜ +15 y ˜ 2 3 ( 6 x ˜ 10 y ˜ +25 y ˜ )   = x ˜ +15 y ˜ 2 3 ( 6 x ˜ +15 y ˜ )   = x ˜ +15 y ˜ +4 x ˜ 10 y ˜   =3 x ˜ +5 y ˜ AE EC = 2( 3 x ˜ +5 y ˜ ) 1( 3 x ˜ +5 y ˜ ) AE:EC=2:1

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