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The distance formula and the Cartesian plane

Let's look at how to find the distance between points P (a, b) and points Q (c, d). In order to find the distance between the points of a right triangle in which we have c - a and d - b, we write the formula under a square root symbol. Furthermore, each part of the equation in parenthese is squared. In other words, we write (c - a)^2 + (d - b)^2 under a square root symbol. This is the equation that represents the distance between the points. But why is this? Because the distance formula is based on the Pythagorean theorem, which teaches that a^2 + b^2 = c^2.

For example, suppose we are trying to find the difference between (2, -6) and (4, 8). We would write (2 - 4)^2 + (-6 - 8)^2. Keep in mind that when the base of an exponent occurs within parentheses, we multiply that number by itself including the negative sign, rather than multiplying the base by itself and then only afterwards adding the negative sign. So within the first set of parentheses, we would write (-2)^2, because 2-4 = -2. This gives us 4. Then we write (-14)^2. This gives us 196. Our final answer is therefore the square root of 200 (though of course, we would have to simplify it if this were a math class).

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