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:

a) The coordinates of the rest stop is [tex]C(x,y) = \left(5, \frac{5}{2} \right)[/tex].

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

Step-by-step 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:

[tex]C(x,y) = \frac{1}{2}\cdot A(x,y) + \frac{1}{2}\cdot B(x,y)[/tex] (1)

Where:

[tex]A(x,y)[/tex] - Coordinates of the high school.

[tex]B(x,y)[/tex] - Coordinates of the stadium.

[tex]C(x,y)[/tex] - Coordinates of the rest stop.

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

[tex]C(x,y) = \frac{1}{2}\cdot (3, 4) + \frac{1}{2}\cdot (7, 1)[/tex]

[tex]C(x,y) = \left(\frac{3}{2}, 2 \right) + \left(\frac{7}{2}, \frac{1}{2} \right)[/tex]

[tex]C(x,y) = \left(5, \frac{5}{2} \right)[/tex]

The coordinates of the rest stop is [tex]C(x,y) = \left(5, \frac{5}{2} \right)[/tex].

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

[tex]d = r\cdot \sqrt{(\Delta x)^{2} + (\Delta y)^{2}}[/tex] (2)

Where:

[tex]r[/tex] - Scale factor, in miles.

[tex]\Delta x[/tex], [tex]\Delta y[/tex] - Horizontal and vertical distances between the high school and the stadium, no unit.

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

[tex]d = r\cdot \sqrt{(\Delta x)^{2} + (\Delta y)^{2}}[/tex]

[tex]d = (6.4\,mi)\cdot \sqrt{4^{2}+(-3)^{2}}[/tex]

[tex]d = 32\,mi[/tex]

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

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