. A treatise on surveying and navigation: uniting the theoretical, the practical, and the educational features of these subjects. m n Whence, AD=-^AB) and BD=—{AB) Now the two parts are numerically known, and are to each other asm to n. Triangles, having the the same altitudes, are to oneanother as their bases. Therefore, ADC : CDB : : m : n as re-quired. Case 2. By a line parallel to one of itssides. Let DE divide the triangle as required,and as similar triangles are to one anotheras the squares of their homologous sides,therefore: (AB)2 : (AD)2 : : m-\-n : n Whence, AD^AB*/ m m-\-n Which sho
. A treatise on surveying and navigation: uniting the theoretical, the practical, and the educational features of these subjects. m n Whence, AD=-^AB) and BD=—{AB) Now the two parts are numerically known, and are to each other asm to n. Triangles, having the the same altitudes, are to oneanother as their bases. Therefore, ADC : CDB : : m : n as re-quired. Case 2. By a line parallel to one of itssides. Let DE divide the triangle as required,and as similar triangles are to one anotheras the squares of their homologous sides,therefore: (AB)2 : (AD)2 : : m-\-n : n Whence, AD^AB*/ m m-\-n Which shows that if we have the numerical value of AB, and ofn and m, we can find that of AD, and from D draw DE parallelto BO, and the triangle is divided as required. Case 3. By a line parallel to a given line, or by a line running ina given direction. To make this case clear, we commence by giving a definite ex-ample : There is a triangular piece of land, from one of the angular points,A, one line runs JV. 25° W., distance 12 chains ; another from the samepoint runs N. 42° E., distance 15 chains. It is required to divide this. DIVISION OF LANDS. 125 triangle into two parts in the ratio 2 to 3, by a line running due eastand west. Let ABC be the given triangle, and BCthe required division line. It is required tofind the numerical value of A C or AB, tomake the area ABC § of the area ABC. Let b represent the side of the triangleopposite B, and c the side opposite C. Let A C=x. As A C and, OB have defi-nite directions, the angle ACB is given,also ABC is given. ACB^%0, ABC=65°, BAC=67°. In the triangle ABC we have
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