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4.1 SLOPE If P1 (x1 , y1 ) and P2 (x2 , y2 ) are two points on a line L , the number m de ned by the equation m= y2 y 1 x2 x 1
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is called the slope of L . The slope measures the steepness of L . It is the ratio of the change y2 y1 in the y-coordinate to the change x2 x1 in the x-coordinate. This is equal to the ratio RP2 /P1 R in Fig. 4-1(a).
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Fig. 4-1 Notice that the value m of the slope does not depend on the pair of points P1 , P2 selected. If another pair P3 (x3 , y3 ) and P4 (x4 , y4 ) is chosen, the same value of m is obtained. In fact, in Fig. 4-1(b), P3 P4 S is similar to P1 P2 R.
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geometry The angles at R and S are both right angles, and the angles at P1 and P3 are equal because they are corresponding angles determined by the line L cutting the parallel lines P1 R and P3 S. Hence, P3 P4 S is similar to P1 P2 R because two angles of the rst triangle are equal to two corresponding angles of the second triangle. 25
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Consequently, since the corresponding sides of similar triangles are proportional, SP4 RP2 = P3 S P1 R or y 2 y1 y4 y3 = x 2 x1 x4 x3
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that is, the slope determined from P1 and P2 is the same as the slope determined from P3 and P4 . EXAMPLE In Fig. 4-2, the slope of the line connecting the points (1, 2) and (3, 5) is
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5 2 3 = = 1.5 3 1 2 Notice that as a point on the line moves two units to the right, it moves three units upward. Observe also that the order in which the points are taken has no effect on the slope: 2 5 3 = = 1.5 1 3 2 In general, y y2 y2 y1 = 1 x2 x1 x1 x2
Fig. 4-2
The slope of a line may be positive, zero, or negative. Let us see what the sign of the slope indicates.
(i) Consider a line L that extends upward as it extends to the right. From Fig. 4-3(a), we see that y2 > y1 ; hence, y2 y1 > 0. In addition, x2 > x1 and, therefore, x2 x1 > 0 . Thus, m= The slope of L is positive. (ii) Consider a line L that extends downward as it extends to the right. From Fig. 4-3(b), we see that y2 < y1 ; therefore, y2 y1 < 0. But x2 > x1 , so x2 x1 > 0. Hence, m= The slope of L is negative. (iii) Consider a horizontal line L . From Fig. 4-3(c), y1 = y2 and, therefore, y2 y1 = 0. Since x2 > x1 , x2 x1 > 0. Hence, m= The slope of L is zero. 0 y2 y1 = =0 x2 x 1 x2 x1 y2 y 1 <0 x2 x 1 y2 y1 >0 x2 x 1
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Fig. 4-3 (iv) Consider a vertical line L . From Fig. 4-3(d), y2 > y1 , so that y2 y1 > 0. But x2 = x1 , so that x2 x1 = 0. Hence, the expression y2 y 1 x2 x1 is unde ned. The concept of slope is not de ned for L . (Sometimes we express this situation by saying that the slope of L is in nite.) Now let us see how the slope varies with the steepness of the line. First let us consider lines with positive slopes, passing through a xed point P1 (x1 , y1 ). One such line is shown in Fig. 4-4. Take another point, P2 (x2 , y2 ) on L such that x2 x1 = 1. Then, by de nition, the slope m is equal to the distance RP2 . Now as the steepness of the line increases, RP2 increases without limit [see Fig. 4-5(a)]. Thus, the slope of L increases from 0 (when L is horizontal) to + when (L is vertical). By a similar construction we can show that as a negatively sloped line becomes steeper and steeper, the slope steadily decreases from 0 (when the line is horizontal) to (when the line is vertical) [see Fig. 4-5(b)].
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