. A treatise on surveying and navigation: uniting the theoretical, the practical, and the educational features of these subjects. Whence, x=a,</ m m-\-n-\-p In the same manner, m-\-n 9—\/i m-f-n-jrp In this manner we might divide the triangle into any proposednumber of parts, having given ratios. Case 3. By lines drawn from a given point on one of the sides ofthe triangle. 132 SURVEYING. Let ABC be the given triangle, and Pthe given point on the side AB. It is required to draw lines from P, asPD and PE, dividing the triangle intothree parts mc, nc,pc, that is, assume thegiven numerical area
. A treatise on surveying and navigation: uniting the theoretical, the practical, and the educational features of these subjects. Whence, x=a,</ m m-\-n-\-p In the same manner, m-\-n 9—\/i m-f-n-jrp In this manner we might divide the triangle into any proposednumber of parts, having given ratios. Case 3. By lines drawn from a given point on one of the sides ofthe triangle. 132 SURVEYING. Let ABC be the given triangle, and Pthe given point on the side AB. It is required to draw lines from P, asPD and PE, dividing the triangle intothree parts mc, nc,pc, that is, assume thegiven numerical area to be {mc-\-nc-\-pc),then the required parts will be mc, nc, andpc*(by Prob. Ill, Mens.) ^SratflSraffii Put AD=xt then •§- ax sin. A=m or x— 2mc a sin. Ac. By comparison, y=- b sin. B When mc and ^>c are cut off, nc is left. Having a and a?, the angle ADP is easily determined. In a similar manner we can divide the triangle into any proposednumber of parts, by lines drawn from the given point 4. By lines drawn from a given point within the triangle. Let AB C be the given triangle, and Pthe given point within it. A variety of lines may be drawn fromP, to divide the triangle into the partsrequired. Conceive PD, PF, and PEto make the requisite division. As P is a given point, AP, PB, and PC are known lines, andthe angles DAP, PAE are known angles. Take AD—b, any convenient assumed value. Take AE= AP=a. Then, \ab sin IL4P=area A DAP Also, \ax sin PAE= area A PAE * Suppose we had a triangular piece of ground containing 320 square rods,and we wished to divide it into three parts in the ratio of 2, 3, and 5, what isthe area of each of the parts ? We decide it thus : m=2. rr=3. p=5. c is at present unknown, but,mc-\-nc-{-pc=320 T
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Keywords: ., boo, bookcentury1800, booksubjectnavigation, booksubjectsurveying
