Question

pq has endpoints at p(-5, 4) and q(7, -5).

1. what is the mid - point of pq?
2. what are the coordinates of the point 2/3 of the way from p to q?
3. what is the length of pq?
4. a chair lift at a ski resort travels along the cable as shown.

1 unit = 10 ft

Explanation:

Step1: Recall mid - point formula

The mid - point formula for two points $(x_1,y_1)$ and $(x_2,y_2)$ is $(\frac{x_1 + x_2}{2},\frac{y_1 + y_2}{2})$. Given $P(-5,4)$ and $Q(7,-5)$, we have $x_1=-5,y_1 = 4,x_2=7,y_2=-5$.

Step2: Calculate x - coordinate of mid - point

$x=\frac{-5 + 7}{2}=\frac{2}{2}=1$.

Step3: Calculate y - coordinate of mid - point

$y=\frac{4+( - 5)}{2}=\frac{4 - 5}{2}=-\frac{1}{2}=-0.5$.
The mid - point of $\overline{PQ}$ is $(1,-0.5)$.

Step4: Recall the formula for a point that divides a line - segment in the ratio $m:n$

The formula for a point $(x,y)$ that divides the line - segment joining $(x_1,y_1)$ and $(x_2,y_2)$ in the ratio $m:n$ is $x=\frac{mx_2+nx_1}{m + n}$ and $y=\frac{my_2+ny_1}{m + n}$. Here, we want the point $\frac{2}{3}$ of the way from $P$ to $Q$, so $m = 2$ and $n = 1$.
$x=\frac{2\times7+1\times(-5)}{2 + 1}=\frac{14 - 5}{3}=\frac{9}{3}=3$.
$y=\frac{2\times(-5)+1\times4}{2 + 1}=\frac{-10 + 4}{3}=\frac{-6}{3}=-2$.
The coordinates of the point $\frac{2}{3}$ of the way from $P$ to $Q$ are $(3,-2)$.

Step5: Recall the distance formula

The distance $d$ between two points $(x_1,y_1)$ and $(x_2,y_2)$ is $d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}$.
For $P(-5,4)$ and $Q(7,-5)$, we have:
$d=\sqrt{(7-(-5))^2+(-5 - 4)^2}=\sqrt{(7 + 5)^2+(-9)^2}=\sqrt{144 + 81}=\sqrt{225}=15$.

Answer:

1. $(1,-0.5)$
2. $(3,-2)$
3. $15$