Machinery's Handbook, 31st Edition
ANALYTIC GEOMETRY 43 has a no slope, since ∆ y = 0. Hence, m = 0/ ∆ x = 0. Between any two points on a vertical line, ∆ x = 0 (Fig. 2d). So, m = ∆ y /0, which is undefined.
Fig. 2. Lines with (a) Positive Slope, m > 0; (b) Negative Slope, m < 0; (c) No Slope, m = 0; (d) Undefined Slope.
Lines and Line Segments.— A line in the plane is the shortest path between two known points, extending indefinitely in both directions. “Line AB ” is notated AB . A line segment is the portion of the line between A and B . The line segment AB is notated AB ; its length is indicated without the bar, as AB . The distinction between the actual line and its length is helpful to keep in mind when referring to the formulas for distance, midpoint, and the other concepts in this section. Distance Between Two Points: The distance d between two points A and B is the length of the line segment connecting them. The formula comes from the Pythagorean theorem, which says that the sum of the squares of the leg measures is the square of the hypotenuse. As labeled in Fig. 3, the legs are lengths x 2 – x 1 and y 2 – y 1 , and the hypotenuse is the distance d between these points. Its formula is: d(A, B) = AB x 2 x 1 – ( ) 2 y 2 y 1 – ( ) 2 + =
Fig. 3. Distance between points A and B . The order in which x and y are subtracted actually does not matter, since the squared difference is the same. Example 1: Find AB, the distance between points A (4, 5) and B (7, 8). Solution: The length of line segment AB is: d 7– 4 ( ) 2 8– 5 ( ) 2 + 3 2 3 2 + 18 3 2 = = = = Midpoint of a Line Segment: The midpoint M(x, y) of line segment AB is found by the coordinate formulas (Fig. 4a): x = x 1 + x 2 and y = y 1 + y 2 2 2 Internal Division of a Line Segment: Point P divides line segment AB (Fig. 4b) in the ratio m:n . That is, P is such a point that AP:PB as m:n . Then, the coordinates of P are given by: x = and y = my 1 + ny 2 mx 1 + nx 2 m + n m + n
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