Question

1.) given p(-5,9) and q(-8,-7), find: a. the distance d(p,q) b. the mid - point of the segment pq c. a point r such that q is the mid - point of pr d. the slope of segment pq

Explanation:

Step1: Recall distance formula

The distance formula 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 points $P(-5,9)$ and $Q(-8,-7)$, $x_1=-5,y_1 = 9,x_2=-8,y_2=-7$.
$d(P,Q)=\sqrt{(-8-(-5))^2+(-7 - 9)^2}$

Step2: Simplify the expression for distance

$d(P,Q)=\sqrt{(-3)^2+(-16)^2}=\sqrt{9 + 256}=\sqrt{265}$

Step3: Recall mid - point formula

The mid - point formula between two points $(x_1,y_1)$ and $(x_2,y_2)$ is $M=(\frac{x_1 + x_2}{2},\frac{y_1 + y_2}{2})$.
For $P(-5,9)$ and $Q(-8,-7)$, the mid - point is $(\frac{-5+( - 8)}{2},\frac{9+( - 7)}{2})=(\frac{-13}{2},1)$

Step4: Use mid - point formula to find $R$

Let $R=(x,y)$. Since $Q(-8,-7)$ is the mid - point of $P(-5,9)$ and $R(x,y)$, we have $\frac{-5 + x}{2}=-8$ and $\frac{9 + y}{2}=-7$.
From $\frac{-5 + x}{2}=-8$, we get $-5+x=-16$, so $x=-11$. From $\frac{9 + y}{2}=-7$, we get $9 + y=-14$, so $y=-23$. So $R=(-11,-23)$

Step5: Recall slope formula

The slope formula between two points $(x_1,y_1)$ and $(x_2,y_2)$ is $m=\frac{y_2 - y_1}{x_2 - x_1}$.
For $P(-5,9)$ and $Q(-8,-7)$, $m=\frac{-7 - 9}{-8-(-5)}=\frac{-16}{-3}=\frac{16}{3}$

Answer:

a. $\sqrt{265}$
b. $(-\frac{13}{2},1)$
c. $(-11,-23)$
d. $\frac{16}{3}$