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

First, let's find the length of line GK using the distance formula:

d = √[(x₂ - x₁)² + (y₂ - y₁)²]
d = √[(8 - 1)² + (12 - 2)²] = √149

Since P is between line GK, then
GK = GP + PK
GK = GP + (3/2)(GP) = √149
GP = 4.882622246

Let's use the distance of GP to find point P(x,y).
4.882622246² = (1 - x)² + (2 - y)²   --> eqn 1

The second equation is the linear equation of line GK using the slope-intercept form.
y = mx + b
Let's use point G(1,2) and point K(8,12)
2 = (12-2)/(8-1)*(1) + b
b = 0.5714285714
So,
y = 10x/7 + 0.5714285714  --> eqn 2

Substitute eqn 2 to substitute eqn 1:
4.882622246² = (1 - x)² + (2 - (10x/7 + 0.5714285714))² 
Solving for x,
x = 3.8
Thus,
y = 10(3.8)/7 + 0.5714285714 = 6

Thus, the coordinates of point P is (3.8,6).