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 }=(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 }=(6-(-10))/(8-(-4))=(6+10)/(8+4)=(16)/(12)=(4)/(3)=1(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)=((x_1+x_2)/(2),(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})_{}=((4)/(2),-(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