203 2.1 Rectangular Coordinates and Graphs To locate on the xy-plane the point corresponding to the ordered pair 13, 42, for example, start at the origin, move to the right 3 units in the positive x-direction, and then move up 4 units in the positive y-direction. See Figure 2. Point A corresponds to the ordered pair 13, 42. The Distance Formula Recall that the distance on a number line between points P and Q with coordinates x1 and x2 is d1P, Q2 = 0 x1 - x2 0 = 0 x2 - x1 0 . Definition of distance By using the coordinates of their ordered pairs, we can extend this idea to find the distance between any two points in a plane. Figure 3 shows the points P1-4, 32 and R18, -22. If we complete a right triangle that has its 90° angle at Q18, 32 as in the figure, the legs have lengths d1P, Q2 = 0 8 - 1-42 0 = 12 and d1Q, R2 = 0 3 - 1-22 0 = 5. By the Pythagorean theorem, the hypotenuse has length 2 122 + 52 = 2144 + 25 = 2169 = 13. Thus, the distance between 1-4, 32 and 18, -22 is 13. x y B(–5, 6) A(3, 4) E(–3, 0) D(4, –3) C(–2, –4) 0 4 units 3 units Figure 2 P(–4, 3) Q(8, 3) R(8, –2) 0 x y Figure 3 0 x d y u y2 – y1u u x2 – x1u R(x2, y2) Q(x2, y1) P(x1, y1) d(P, R) = !(x2 – x1) 2 + (y 2 – y1) 2 Figure 4 To obtain a general formula, let P1x1, y12 and R1x2, y22 be any two distinct points in a plane, as shown in Figure 4. Complete a triangle by locating point Q with coordinates 1x2, y12. The Pythagorean theorem gives the distance between P and R. d1P, R2 = 21x2 - x122 + 1y 2 - y122 Absolute value bars are not necessary in this formula because, for all real numbers a and b, 0 a - b0 2 = 1a - b22. The distance formula can be summarized as follows. René Descartes (1596–1650) The initial flash of analytic geometry may have come to Descartes as he was watching a fly crawling about on the ceiling near a corner of his room. It struck him that the path of the fly on the ceiling could be described if only one knew the relation connecting the fly’s distances from two adjacent walls. Locating a fly on a ceiling Distance Formula Suppose that P1x1, y12 and R1x2, y22 are two points in a coordinate plane. The distance between P and R, written d1P, R2, is given by the following formula. d1P, R2 =!1x2 −x12 2 + 1 y 2 −y12 2 That is, the distance between two points in a coordinate plane is the square root of the sum of the square of the difference of their x-coordinates and the square of the difference of their y-coordinates. Data from An Introduction to the History of Mathematics by Howard Eves.
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