In the trapezoid abcd (
ab
∥
cd
) point m∈
ad
, so that am: md=3: 5. line l ∥
ab
and going trough point m intersects diagonal
ac
and leg
bc
at points p and n respectively. find:

ap: pc and bn: cn

  AP:PC = BN:CN = 3:5

Step-by-step explanation:

The horizontal line MN divides any line between AB and CD into the same ratio as AM:MD, which is 3:5.

  AP:PC = BN:CN = 3:5

_____

Since MP is parallel to AC, triangles AMP and ADC are similar with the scale factor AM:AD = 3:8. Likewise, triangles CNP and CBA are similar with the scale factor 5:8. These similarities and scale factors make ...

  (AP+PC)/AP = 8/3

  1 +PC/AP = 1 +5/3 . . . . . rewrite

  AP/PC = 3/5 . . . . . . . . .subtract 1 and invert

__

  (CN+BN)/CN = 8/5

  1 +BN/CN = 1 +3/5 . . . . . rewrite

  BN/CN = 3/5 . . . . . . . . . subtract 1

the answer for this question is 38

set each factor equal to 0

4x-3 = 0

2x-1 = 0

solve for x.

4x-3=0

4x=3

x= 3/4

solve for x. 

2x-1= 0

2x= 1

x= 1/2

eventually after a few more steps, your answer should be 1/2 is less than or equal to x and 3/4 is greater than or equal to x