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Straight Lines in .NET framework
Straight Lines Decode Quick Response Code In .NET Using Barcode Control SDK for .NET framework Control to generate, create, read, scan barcode image in .NET applications. QR Code JIS X 0510 Maker In .NET Framework Using Barcode maker for Visual Studio .NET Control to generate, create Denso QR Bar Code image in VS .NET applications. 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 QR Code Recognizer In VS .NET Using Barcode reader for .NET Control to read, scan read, scan image in VS .NET applications. Making Barcode In Visual Studio .NET Using Barcode encoder for .NET framework Control to generate, create barcode image in .NET framework applications. is called the slope of L . The slope measures the steepness of L . It is the ratio of the change y2 y1 in the ycoordinate to the change x2 x1 in the xcoordinate. This is equal to the ratio RP2 /P1 R in Fig. 41(a). Barcode Reader In VS .NET Using Barcode reader for VS .NET Control to read, scan read, scan image in Visual Studio .NET applications. QR Code Maker In Visual C#.NET Using Barcode creator for .NET Control to generate, create QRCode image in .NET applications. Fig. 41 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. 41(b), P3 P4 S is similar to P1 P2 R. Generating QR Code In Visual Studio .NET Using Barcode printer for ASP.NET Control to generate, create QRCode image in ASP.NET applications. QR Creator In Visual Basic .NET Using Barcode encoder for .NET framework Control to generate, create QR image in .NET framework applications. 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 Create Matrix 2D Barcode In .NET Using Barcode generation for VS .NET Control to generate, create Matrix Barcode image in .NET framework applications. Linear 1D Barcode Generator In .NET Framework Using Barcode creator for .NET framework Control to generate, create Linear 1D Barcode image in VS .NET applications. Copyright 2008, 1997, 1985 by The McGrawHill Companies, Inc. Click here for terms of use.
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Painting EAN13 In .NET Using Barcode printer for Reporting Service Control to generate, create EAN13 image in Reporting Service applications. Code128 Creation In Java Using Barcode creator for BIRT reports Control to generate, create USS Code 128 image in BIRT reports applications. [CHAP. 4
Code 39 Extended Creator In .NET Using Barcode encoder for Reporting Service Control to generate, create USS Code 39 image in Reporting Service applications. Encoding GTIN  12 In .NET Using Barcode creator for Reporting Service Control to generate, create UPCA Supplement 2 image in Reporting Service applications. 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 Code 128C Drawer In None Using Barcode generator for Office Word Control to generate, create Code128 image in Microsoft Word applications. EAN13 Recognizer In None Using Barcode scanner for Software Control to read, scan read, scan image in Software applications. that is, the slope determined from P1 and P2 is the same as the slope determined from P3 and P4 . EXAMPLE In Fig. 42, the slope of the line connecting the points (1, 2) and (3, 5) is Print UPCA Supplement 2 In Java Using Barcode drawer for Java Control to generate, create UPCA image in Java applications. UCC128 Printer In None Using Barcode creator for Font Control to generate, create EAN128 image in Font applications. 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. 42 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. 43(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. 43(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. 43(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 CHAP. 4] STRAIGHT LINES
Fig. 43 (iv) Consider a vertical line L . From Fig. 43(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. 44. 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. 45(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. 45(b)].

