SOLUTION: rectangle ABCD has AB = 16 and BC = 6. Let M be the midpoint of side AD and N be the midpoint of side CD. Segments CM and An intersect at G. Find the length AG. I did this on th

Algebra ->  Rectangles -> SOLUTION: rectangle ABCD has AB = 16 and BC = 6. Let M be the midpoint of side AD and N be the midpoint of side CD. Segments CM and An intersect at G. Find the length AG. I did this on th      Log On


   

Question 148149: rectangle ABCD has AB = 16 and BC = 6. Let M be the midpoint of side AD and N be the midpoint of side CD. Segments CM and An intersect at G. Find the length AG.
I did this on the geometer's sketchpad and i got 6.66748, but i don't understand how to do it by hand. Thanks for the help!!!
Answer by scott8148(6628) About Me  (Show Source):
You can put this solution on YOUR website!
put the rectangle on a coordinate system

D is the origin (0,0) __ AD runs up the y-axis and DC goes along the x-axis

the equation for AN is y=(-3/4)x+6 __ the equation for CM is y=(-3/16)x+3

for G (the intersection) (-3/4)x+6=(-3/16)x+3 __ adding 3x/4 and -3 __ 3=(9/16)x __ dividing by 9/16 __ 16/3=x

substituting to find y __ y=(-3/16)(16/3)+3 __ y=-1+3 __ y=2

so the coordinates of G are (16/3,2)

using the distance formula __ (AG)^2=(16/3)^2 + (6-2)^2 __ AG^2=256/9+16 __ AG^2=256/9+144/9 __ AG^2=400/9 __ AG=20/3