ACGC - Distance and Midpoint (Lesson)

Distance and Midpoint 

The Distance Formula

We can use the coordinates of a line to find the distance of the line. Watch this video and work along with the example.

Distance Formula  = $\sqrt[]{\left(x_2-x_1\right)^2+\left(y_2-y_1\right)^2}^{ }$

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Application: Distance

Example: Determine the length of the creek if each square in the grid represents 10 feet.

Use the distance formula and the points (0, 2) and (10, 8)

$\sqrt[]{\left(x_2-x_1\right)^2+\left(y_2-y_1\right)^2}^{ }$

$\sqrt{\left(10-0\right)^2+\left(8-2\right)^2}$

$\sqrt{\left(10\right)^2+\left(6\right)^2}$

$\sqrt{100+36}$

$\sqrt{136}\approx11.66$ 

Since each square represents 10 feet, the creek is 11.66 x 10 = 116.6 ft

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The Midpoint Formula

We can also use the coordinates of a line to find the midpoint of the line. 

Example: Find the midpoint of a line with endpoints (1, 9) and (-5, 10).

Midpoint Formula = $(\frac{x_2+x_1}{2},\frac{y_2+y_1}{2})$

$(\frac{1+-5}{2},\frac{9+10}{2})$              Substitute in the coordinates

$(\frac{-4}{2},\frac{19}{2})$                      Simplify

$(-2,9.5)$                     Simplify

The midpoint of this line is $(-2,9.5)$.

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Application: Midpoint

Example: The next phase of our plan includes placing a water fountain at the halfway point of the creek. Use the midpoint formula to find the halfway point of the creek.

Use the midpoint formula and the points (0, 2) and (10, 8): 

$(\frac{x_2+x_1}{2},\frac{y_2+y_1}{2})$

$(\frac{10+0}{2},\frac{8+2}{2})$                Substitute in the coordinates

$(\frac{10}{2},\frac{10}{2})$                      Simplify

$(5,5)$                            Simplify

The midpoint of the creek is $(5,5)$.

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