Exercise 6 Page 160

A ratio of $2$ to $6$ indicates that for every $2$ of one thing, $6$ of another thing will occur. To compare the lengths of the two line segments, $\overline{AP}$ and $\overline{PB},$ let's say that they are made of units of length $x.$

We see here that $\overline{AP}$ is of length $2x,$ while $\overline{PB}$ is of length $6x,$ thus fulfilling our ratio of $2:6.$ If we combine these two segments into one collinear segment, we can form the segment $\overline{AB}.$

Therefore, the point $P$ is $2$ out of $8$ steps between $A$ and $B.$ Now let's draw the segment $\overline{AB}$ in a coordinate plane using the given coordinates for $A(\text{-}3,2)$ and $B(5,\text{-}4).$

To move from $A$ to $P,$ we need to move $\frac{2}{8}\text{ths}$ of the way from $A$ to $B$ in both the vertical and horizontal directions. To calculate this, we need to find the segment's slope. Remember to leave the slope in terms of rise and run and do not simplify. Knowing the slope, we can write equations for finding the $x\text{-}$ and $y\text{-}$coordinates of $P$ using the rise and run separately. We have to move $2$ steps in the positive horizontal direction and $1.5$ steps in the negative vertical direction to move from point $A$ to point $P.$