Given point a with coordinates (3, 5) and point b with coordinates (12, 9)… • find point c that splits segment ab into a ratio of 4: 7 • find point d that is located 2/5 of the way from point a to point b

Refer to the diagram shown below.

Part 1
Let the distance AC = 4d and distance BC = 7d so that C splits Ab in a 4:7 ratio.
Then
(12 - x)/(x - 4) = 7/4
Cross multiply.
7x - 28 = 48 - 4x
11x = 76
x = 6.909
Similarly,
(9 - y)/(y - 5) = 7/4
7y - 35 = 36 - 4y
11y = 71
y = 6.454

The coordinates of C are (6.909, 6.454).

Part 2
Let AD = 2d and BD = 5d so that D s located 2/5 from A to B.
Then
(x - 3)/(12 - x) = 2/3
Cross multiply.
3x - 9 = 24 - 2x
5x = 33
x =  6.6
Similarly,
(y - 5)/(9 - y) = 2/3
3y - 15 = 18 - 2y
5y = 33
y = 6.6

The coordinates of D are (6.6, 6.6)