205 2.1 Rectangular Coordinates and Graphs P R Q d(P, R) d(P, Q) d(Q, R) d(P, Q) + d(Q, R) = d(P, R) Figure 6 Using a similar procedure, we can determine whether three points are collinear (that is, lying on a straight line). See Figure 6. Three points are collinear if the sum of the distances between two pairs of the points is equal to the distance between the remaining pair of points. EXAMPLE 4 Applying the Distance Formula Determine whether the points P1-1, 52, Q12, -42, and R14, -102 are collinear. SOLUTION Use the distance formula. d1P, Q2 = 21-1 - 222 + 35 - 1-4242 = 29 + 81 = 290 = 3210 290 = 29 # 10 = 3210 d1Q, R2 = 212 - 422 + 3-4 - 1-10242 = 24 + 36 = 240 = 2210 240 = 24 # 10 = 2210 d1P, R2 = 21-1 - 422 + 35 - 1-10242 = 225 + 225 = 2250 = 5210 2250 = 225 # 10 = 5210 Because 3210 + 2210 = 5210 is true, the three points are collinear. S Now Try Exercise 29. The Midpoint Formula The midpoint of a line segment is equidistant from the endpoints of the segment. The midpoint formula is used to find the coordinates of the midpoint of a line segment. To develop the midpoint formula, let P1x1, y12 and Q1x2, y22 be any two distinct points in a plane. (Although Figure 7 shows x1 6x2, no particular order is required.) Let M1x, y2 be the midpoint of the segment joining P and Q. Draw vertical lines from each of the three points to the x-axis, as shown in Figure 7. The ordered pair M1x, y2 is the midpoint of the line segment joining P and Q, so the distance between x and x1 equals the distance between x and x2. x2 - x = x - x1 x2 + x1 = 2x Add x and x1 to each side. x = x1 + x2 2 Divide by 2 and rewrite. Similarly, the y-coordinate is y1 + y2 2 . Q(x2, y2) P(x1, y1) y x x2 x x1 0 M(x, y) Figure 7 Midpoint Formula The coordinates of the midpoint M of the line segment with endpoints P1x1, y12 and Q1x2, y22 are given by the following formula. M= a x1 +x2 2 , y1 +y2 2 b That is, the x-coordinate of the midpoint of a line segment is the average of the x-coordinates of the segment’s endpoints, and the y-coordinate is the average of the y-coordinates of the segment’s endpoints. Solve for x.
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