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# help with analytic geometry

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Let $A = (0, 0),$ $B = (1, 2),$ $C=(3, 3),$ and $D = (4, 0).$ Quadrilateral $ABCD$ is cut into two pieces with the same area by a line passing through $A$. What are the coordinates of the point where this line intersects $\overline{CD}$?

Mar 10, 2021

#1
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We can quickly figure out that the line through A must have slope 1/2.  So y = x/2, and the line CD is y = -3x + 12.  Solving these equations, you can find (x,y) = (24/7,12/7).

Mar 10, 2021
#2
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Really close it was (25/8, 15/8) but thanks for the help :)

Mar 10, 2021
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x = 27/8

jugoslav  Mar 10, 2021
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[ABC] = 1.5        [ADN] = 1.5

EN = (1.5 * 2) / 4 = 0.75

MF = ((3 - 0.75)/2) + 0.75 = 1.875

DF = 1/3(1.875) = 0.625

AF = AD - DF = 3.375

Coordinates  M (3 3/8, 1 7/8 )        ( M is the midpoint on the line segment CN)

Mar 10, 2021