Given g (1,2) and k (8,12) what does it mean to find the point p on gk such that 3 (gp) = 2(pk)

It means you want to find the point along the line GK that divides the length of the line into a 2/3 ratio.



GK = GP+PK = 2+3 = 5

In this way, we can think of the line as being in 5 equal parts, 2 parts consist of GP, and 3 parts for PK.

Now find the horizontal and vertical distances from point g to k.
H: 8 - 1 = 7
V: 12 - 2 = 10

Divide by 5
H: 7/5
V: 10/5 = 2

Adding these to point 'g' will get you to some point that is 1/5 the length of GK.
But point 'p' must be 2 parts away from 'g'.

Multiply by 2
H: 7/5*2 = 14/5
V: 2*2 = 4

Add to 'g'
p = (1+14/5, 2 +4)

Ac equals 35 cm long