1. ## Line Segments

Hi guys, I have an exam tomorrow about line segments, but I can't seem to understand how to find the Constant(K) or the Ratio and Proportion method. Can anyone explain this to me please?

For example

Find the trisection points of line segment joining (-2,5) and (7,1)

How do I find the value of K? and also R1/R2

Another one, Line segment (-2,-3) to (6,1) is extended to a distance of 1/4 its original length. What is K? R1? R2?

Thank you!

2. Let $A(x_1,y_1), \ B(x_2,y_2), \ M(x,y)$ such as $\frac{AM}{MB}=k$

Then

$\left\{\begin{array}{ll}x=\frac{x_1+kx_2}{1+k}\\y= \frac{y_1+ky_2}{1+k}\end{array}\right.$

In this case $A(-2,5), \ B(7,1), \ M(a,b), \ N(c,d)$

and $\frac{AM}{MB}=\frac{1}{2}, \ \frac{AN}{NB}=2$

Can you continue?