+0

0
10
2
+214

ABCDEF is a regular hexagon. M and N are the midpoints of DE and AF. P is the intersection of two lines BM and DN. What is the ratio of DP to PN? Express your answer as a common fraction.

#1
+128794
+1

Coordinate Geometry

Let the center  of the hexagon = (0,0)

Let ED  = 2

D= (1, sqrt 3)

F = (-2,0)

A= (-1 , -sqrt 3)

N =  (-3/2 , -sqrt (3) / 2 )

M = (0, sqrt (3) )

B = (1 , -sqrt (3))

Slope of line containing  segment  (DN) =    [ sqrt (3) + sqrt (3) / 2 ] / [5/2]  =  3sqrt (3) / 5

Equation of this  line

y = [ (3/5)sqrt 3 ]  ( x - 1) + sqrt (3)         (1)

Slope of line  containing segment  MB  =  [ 2sqrt (3) ] / -1 =  -2sqrt (3)

Equation of this  line

y = [ -2sqrt (3) ] (x ) + sqrt (3)         (2)

Set  (1) = (2)   to find the x coordinate of P

[ (3/5) sqrt (3) ] (x -1) + sqrt (3)  =  [-2sqrt (3)]x + sqrt (3)

(3/5)sqrt (3) x - (3/5)sqrt (3) = -2sqrt (3)x

(13/5) sqrt (3) x  =  (3/5)sqrt (3)

(13/5)x = (3/5)

x = (3/5)(5/13)  = 15/65  =  3/13

y =  -2sqrt (3) (3/13) + sqrt (3)

y =  (7/13)sqrt (3)

DP^2  = ( 1-3/13)^2 + ( sqrt 3 - (7/13)sqrt 3)^2

DP^2  = (10/13)^2  + [(6/13) sqrt 3]^2

DP^2  =  16/13

PN^2  = ( 3/13 + 3/2)^2 + [ (7/13)sqrt 3  + sqrt (3)  / 2)^2

PN^2   = (45/26)^2  +  (27sqrt (3) / 26)^2

PN^2  =   81/13

DP^2 / PN^2  =   16 / 81

DP  / PN  =   4 / 9

Apr 7, 2024
#2
+214
+1

Thanks so much