Question

A high school soccer team is going to Columbus to see a professional soccer game. A coordinate grid is superimposed on a highway map of Ohio. The high school is at point (3, 4) and the stadium in Columbus is at point (7, 1). The map shows a highway rest stop halfway between the cities. What are the coordinates of the rest stop? What is the approximate distance between the high school and the stadium? (One unit = 6.4 miles.)

Answer 1

a) The coordinates of the rest stop is
`C(x,y) = \left(5, (5)/(2) \right)`.

b) The approximate distance between the high school and the stadium is 32 miles.

Explanation:

a) The rest stop is located in the midpoint of the line segment between the high school and the stadium. Vectorially speaking, we use the following formula:

`C(x,y) = (1)/(2)\cdot A(x,y) + (1)/(2)\cdot B(x,y)` (1)

Where:

`A(x,y)` - Coordinates of the high school.

`B(x,y)` - Coordinates of the stadium.

`C(x,y)` - Coordinates of the rest stop.

If we know that
`A(x,y) = (3, 4)` and
`B(x,y) = (7,1)`, then the coordinates of the rest stop are, respectively:

`C(x,y) = (1)/(2)\cdot (3, 4) + (1)/(2)\cdot (7, 1)`

`C(x,y) = \left((3)/(2), 2 \right) + \left((7)/(2), (1)/(2) \right)`

`C(x,y) = \left(5, (5)/(2) \right)`

The coordinates of the rest stop is
`C(x,y) = \left(5, (5)/(2) \right)`.

b) The approximate distance between the high school and the stadium (
`d`), in miles, is the product of the Length Equation of the Line Segment and the scale factor:

`d = r\cdot \sqrt{(\Delta x)^(2) + (\Delta y)^(2)}` (2)

Where:

`r` - Scale factor, in miles.

`\Delta x`,
`\Delta y` - Horizontal and vertical distances between the high school and the stadium, no unit.

If we know that
`r = 6.4\,mi`,
`\Delta x = 4` and
`\Delta y = -3`, then the distance between the high school and the stadium is:

`d = r\cdot \sqrt{(\Delta x)^(2) + (\Delta y)^(2)}`

`d = (6.4\,mi)\cdot \sqrt{4^(2)+(-3)^(2)}`

`d = 32\,mi`

The approximate distance between the high school and the stadium is 32 miles.