Question

The planners in the local department of transportation want to build a

highway with two rest stops between the towns. The rest stops will divide
the highway into three equal parts.
Where, on the coordinate plane, should the rest stops should be built?

Answer 1

(-0.75, -3.25) and (4.125, -2.125)

Explanation:

Let the town, Thayer = point A(-4, -4)

Town Purdy = point D(9, -1).

Let B and C represent the two rest stops that divides the highway (AD) into 3 equal parts, namely: AB = BC = CD.

Distance of point B from A to D = ⅓ of AD.

Point C = the midpoint of point B and D.

First, using the formula for internal division, let's find the coordinates of point B:

Internal division to find the coordinates of point B is given as:

`x = (mx_2 + nx_1)/(m + n)`

`y = (my_2 + ny_1)/(m + n)`

Where,

`A(-4, -4) = (x_1, y_1)`

`D(9, -1) = (x_2, y_2)`

`m = 1, n = 3`

Plug in the necessary values into the formula stated above to solve for x and y of point B:

`x = (1*9 + 3*(-4))/(1 + 3)`

`x = (9 - 12)/(4)`

`x = (-3)/(4)`

`x = -(-3)/(4) = 0.75`

`y = (1*(-1) + 3*(-4))/(m + n)`

`y = (-1 - 12)/(1 + 3)`

`y = (-13)/(4)`

`y = -3.25`

The coordinates of point B = (-0.75, -3.25)

Next, find the coordinates of point C, which is the midpoint of B(-0.75, -3.25) and D(9, -1) using the midpoint formula.

Midpoint (C) of BC, for B(-0.75, -3.25) and D(9, -1) is given as:

`M((x_1 + x_2)/(2), (y_1 + y_2)/(2))`

Let
`B(-0.75, -3.25) = (x_1, y_1)`

`D(9, -1) = (x_2, y_2)`

Thus:

`C((-0.75 + 9)/(2), (-3.25 + (-1))/(2))`

`C((8.25)/(2), (-4.25)/(2))`

`C(4.125, -2.125)`

The rest stops should be built at (-0.75, -3.25) and (4.125, -2.125) in the coordinate plane.