Answer to Question #295363 in Geometry for Emiey

Question

D and E are points on sides of AB and BC respectively of ∆ABC such that AD:DB= 2:3 and BE=4÷3 EC. If DK||AE and AE and CD intersect at P, Find the ratio of CP:PD

Answer 1

SOLUTION:

Let AB and AC be vectors b and c respectively

let DP= kDC and AP = mAE

AP=m(b+3÷7BC)

BC= -b+c

AP= b+ 3÷7c - 3÷7b

= (3÷7c - 4÷7b)m

= 3÷7mc - 4÷7mb

AD+DP

2÷7b +p( -2÷3b +c)

(2÷5-2÷5p)b + PC

Comparing the coefficients of a and b we take

3÷7m = p

2÷5-2÷5k = 4÷7m

(2÷5 - 2÷5) × (3÷7)m = 4÷7m

2÷5 - 6÷35m = 4÷7m

2÷5 = 26÷35m = 7÷15

Therefore, k = 3÷7 × 7÷15

k = 1÷5 and (k-1) = 4÷5

The ratio of CP:PD = 1:4