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In the diagram, D and  E are the midpoints of AB and BC respectively. Determine the sum of the X and Y coordinates of F, the point of intersection of AE and CD

 

 

 

 

 

 

 

https://latex.artofproblemsolving.com/5/5/0/55002ffdc5d288b4984c4eba461a467dcced7760.png

 Aug 21, 2020
 #1
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First, lets find the coordinates of D and E.

 

D is the midpoint of A(0,6) and B(0,0)

E is the midpoint of B(0,0) and C(8,0)

 

The coordinates of D and E are (0,3) and (4,0), respectively.

 

Since we know the coords of D and E, we can now find the equation for lines AE and CD.

 

Line AE is a line that passes through the points (0,6) and (4,0).

The slope of AE is (0-6)/(4-0) = -6/4 = -3/2

So AE is a line that has a slope of -3/2 passing through the points (0,6) and (4,0)

 

y = (-3/2)x + b

6 = 0 + b, 

b = 6

 

y = (-3/2)x + 6

 

Do the same process of finding the slope for line CD and plugging in coords to get the equation for line CD:

 

y = (-3/8)x + 3

 

Since F is the intersection between the two lines, all we have to do is put the two equations equal to each other and solve.

 

(-3/2)x + 6 = (-3/8)x + 3

x = 8/3

 

Plug that x value into any y to get

y = 2

 

And from there just add 2 and 8/3 to get:

 

14/3 or 4.66...

 Aug 21, 2020

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