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The distance and midpoint formulas

The distance formula is used to find the distance between two points in the coordinate plane. We'll explain this using an example below

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We want to calculate the distance between the two points (-2, 1) and (4, 3). We could see the line drawn between these two points is the hypotenuse of a right triangle. The legs of this triangle would be parallel to the axes which mean that we can measure the length of the legs easily.

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We'll get the length of the distance d by using the Pythagorean Theorem

\\ d^{2}=2^{2}+6^{2}=4+36=40 \\\\d=\sqrt{40}\approx 6.32 \\

This method can be used to determine the distance between any two points in a coordinate plane and is summarized in the distance formula

\\ d=\sqrt{\left ( x_{2}-x_{1} \right )^{2}+\left ( y_{2}-y_{1} \right )^{2}} \\

The point that is at the same distance from two points A (x1, y1) and B (x2, y2) on a line is called the midpoint. You calculate the midpoint using the midpoint formula

\\ m =\left ( \frac{x_{1}+x_{2}}{2} \right ),\: \: \left ( \frac{y_{1}+y_{2}}{2} \right ) \\

We can use the example above to illustrate this

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\\ m =\left ( \frac{4+(-2)}{2} \right ),\: \: \left ( \frac{3+1}{2} \right )=\\\\\\=\left ( \frac{2}{2} \right ),\: \: \left ( \frac{4}{2} \right )=\begin{pmatrix} 1,\: 2 \end{pmatrix} \\

Video lesson: Calculate the distance between the two points

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Next Class:  Rational expressions, Simplify rational expression