. A treatise on surveying and navigation: uniting the theoretical, the practical, and the educational features of these subjects. om a given point in the perimeter. Let ABCD be the given quadri-lateral, and P the given point in theside AB. It is required to determine themagnitude, and the direction of the line PG, which divides thefigure into parts in the ratio of m to n. All the sides and anglesof the quadrilateral are known, and its area is known. AB andDOr are, or are not parallel; if they are parallel the figure is a tra-pezoid, then the method of finding G, in the opposite side, iseasy an
. A treatise on surveying and navigation: uniting the theoretical, the practical, and the educational features of these subjects. om a given point in the perimeter. Let ABCD be the given quadri-lateral, and P the given point in theside AB. It is required to determine themagnitude, and the direction of the line PG, which divides thefigure into parts in the ratio of m to n. All the sides and anglesof the quadrilateral are known, and its area is known. AB andDOr are, or are not parallel; if they are parallel the figure is a tra-pezoid, then the method of finding G, in the opposite side, iseasy and obvious. If AB and CD are not parallel, we can pro-duce them and form a triangle in the one direction or the other; bythis figure we form the triangle BCE, whose area we may repre-sent by t. As B C, and the angles as B and C are all known, the triangleEB C is determined in all respects. As PB is known, PE is known, and designate EG by x. Letcm and en designate the portions of the quadrilateral after it is di-vided, and let cm represent the part BPGC. Put PE=a. Now ( by Problem III, Mensuration ), we have\ax sin. E=t-\ DIVISION OF LANDS. 135 2t-\-2cm Whence, x=zcT^rE We now have the numerical value of x, from which we subtractEC, and we have CG, which being measured from C will give thepoint G, through which to draw the line from P, to divide thefigure as required. Case 2. By a line making a given angle with one of the sides. If the division line makes a given angle with one side, it mustalso make a known angle with the opposite side. Taking the last figure, conceiving PG to take a given directionacross AB and CD, so as to cut off the area mc. As in the former case; let t represent the area of the triangle EB C,to this add mc, and we have the area of the triangle P GE. But inthis case P is not a given point, and EP is not known. Put EG=x. Let P represent the given angle at P, and G thegiven angle at G. Now, by trigonometry, sin. P : x : : sin. G : EP Or, EP=^®x sin. P (P
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Keywords: ., boo, bookcentury1800, booksubjectnavigation, booksubjectsurveying
