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=10x˜, →AD=25y˜ and →BC=−x˜+15y˜.(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 =25y˜−10x˜
(a)(ii)
→AE=→AB+→BE =→AB+25→BD =10x˜+25(25y˜−10x˜) =10x˜+25(25y˜−10x˜) =10x˜+10y˜−4x˜ =6x˜+10y˜ =2(3x˜+5y˜)
(b)
→EC=→EB+→BC =→BC−→BE =→BC−23→ED =→BC−23(→EA+→AD) =−x˜+15y˜−23(−6x˜−10y˜+25y˜) =−x˜+15y˜−23(−6x˜+15y˜) =−x˜+15y˜+4x˜−10y˜ =3x˜+5y˜AEEC=2(3x˜+5y˜)1(3x˜+5y˜)AE:EC=2:1
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=10x˜, →AD=25y˜ and →BC=−x˜+15y˜.(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 =25y˜−10x˜
(a)(ii)
→AE=→AB+→BE =→AB+25→BD =10x˜+25(25y˜−10x˜) =10x˜+25(25y˜−10x˜) =10x˜+10y˜−4x˜ =6x˜+10y˜ =2(3x˜+5y˜)
(b)
→EC=→EB+→BC =→BC−→BE =→BC−23→ED =→BC−23(→EA+→AD) =−x˜+15y˜−23(−6x˜−10y˜+25y˜) =−x˜+15y˜−23(−6x˜+15y˜) =−x˜+15y˜+4x˜−10y˜ =3x˜+5y˜AEEC=2(3x˜+5y˜)1(3x˜+5y˜)AE:EC=2:1