Unformatted text preview: now is to create an algebraic formula to compute the distance between these two
points. Consider the generic situation below on the left.
Q (x2 , y2 ) Q (x2 , y2 ) d P (x1 , y1 ) d P (x1 , y1 ) (x2 , y1 ) With a little more imagination, we can envision a right triangle whose hypotenuse has length d as
drawn above on the right. From the latter ﬁgure, we see that the lengths of the legs of the triangle
are |x2 − x1 | and |y2 − y1 | so the Pythagorean Theorem gives us
|x2 − x1 |2 + |y2 − y1 |2 = d2
(x2 − x1 )2 + (y2 − y1 )2 = d2
(Do you remember why we can replace the absolute value notation with parentheses?) By extracting
the square root of both sides of the second equation and using the fact that distance is never
negative, we get 1.1 The Cartesian Coordinate Plane 7 Equation 1.1. The Distance Formula: The distance d between the points P (x1 , y1 ) and
Q (x2 , y2 ) is:
(x2 − x1 )2 + (y2 − y1 )2 d= It is not always the case that the points P and Q lend themselves to constructing such a triangle.
If the points P and...
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