D is the midpoint of A and B, so it has coordinates
D ((0 +0)/2, (9 + 0)/2) = D (0, 9/2)
and E is the midpoint of B and C, so its coordinates are
E ((0 + 13)/2, (0 + 0)/2) = E (13/2, 0)
(a) The slope of the line through C and D is
(9/2 - 0)/(0 - 13) = -9/26
Then the equation of the line through C and D is
y - 0 = -9/26 (x - 13)
y = -9/26 x + 9/2
(b) I assume this part is asking about the coordinates of F as the intersection of AE and CD.
First find the equation of the line through A and E. It has slope
(9 - 0)/(0 - 13/2) = -18/13
so its equation is
y - 0 = -18/13 (x - 13/2)
y = -18/13 x + 9
Solve for the x-coordinate of F:
-9/26 x + 9/2 = -18/13 x + 9
(-9/26 + 18/13) x = 9 - 9/2
(36 - 9)/26 x = 9/2
27/26 x = 9/2
x = (9/2)/(27/26)
x = 13/3
Then use either line equation to solve for the y-coordinate:
y = -9/26 (13/3) + 9/2
y = -39/26 + 9/2
y = (117 - 39)/26
y = 78/26
y = 3
So F is the point (13/3, 3).
(c) Triangle ABC has height 9 and length 13, so its area is
1/2 • 9 • 13 = 117/2
(d) Triangle BCD has area
1/2 • 9/2 • 13 = 117/4
Triangle ECF has height 3 (the y-coordinate of F) and length 13 - 13/2 = 13/2 (the difference between the x-coordinate of C and E), so its area is
1/2 • 3 • 13/2 = 39/4
The area of BCD is the sum of the areas of quadrilateral DBEF and triangle ECF. So the area of DBEF is
117/4 - 39/4 = (117 - 39)/4 = 78/4 = 39/2
