Question

For each pair below, find three quantities: the slope between the points, the midpoint between the points and the distance between the points. Show all calculations. Simplify all answers.

Answer 1

Step 1: Write out the formula for the slope of the line joining two points

$\begin{gathered} \text{ Slope }=\frac{y_2-y_1}{x_2-x_1} \\ \text{ Where} \\ (x_1,y_1)\text{ is the first point and} \\ (x_2,y_2)\text{ is the second point} \end{gathered}$

Step 2: Find the slope of the line joining points A and B

We can set (x1,y1) = (-4, -10) and (x2,y2) = (8,6)

Therefore,

$\text{Slope }=\frac{6-(-10)}{8-(-4)}=\frac{6+10}{8+4}=\frac{16}{12}=\frac{4}{3}=1\frac{1}{3}$

Hence the slope is 1 1/3

Step 3: Write out the formula for finding the midpoint (xm,ym) of two points (x1,y1) and (x2,y2)

$(x_m,y_m)=(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2})_{}$

Step 4: Find the midpoint between points A and B

We can set (x1,y1) = (-4, -10) and (x2,y2) = (8,6)

$(x_m,y_m)=(\frac{-4_{}+8_{}}{2},\frac{-10_{}+6_{}}{2})_{}=(\frac{4}{2},-\frac{4}{2})=(2,-2)$

The midpoint of A(-4, 10) and B(8,6) is (2, -2)

Step 4: Write out the formula for finding the distance between two points (x1,y1) and (x2,y2)

$d=\sqrt[]{(x_{2-}x_1)^2_{}+(y_2-y_1)^2}$$\begin{gathered} \text{ Where} \\ d=\text{ the distance betwe}en\text{ the points} \\ (x_1,y_1)\text{ the first point} \\ (x_2,y_2)\text{ is the second point} \end{gathered}$

Step 5: Find the distance between points A and B

We can set (x1,y1) = (-4, -10) and (x2,y2) = (8,6)

$\begin{gathered} d=\sqrt[]{(8-(-4))^2+(6-(-10))^2}=\sqrt[]{(8+4)^2+(6+10)^2}=\sqrt[]{12^2+16^2}=\sqrt[]{144+256}=\sqrt[]{400} \\ d=20\text{ units} \end{gathered}$

Hence the distance between points A and B is 20 units