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In the right angled Triangle ABC, the base
BC and the sum of the perpendicular and fides, B.4 +AC+AD, being given, to find the Triangle.
Such parts of this triangle are to be found as are necessary for defcribing it: The perpendicularAD C will be sufficient
D for this purpose, and let it be called x: Let AB + AC + AD = a, BC=b, therefore BA+AC=a-x: Let BA-AC be denoted by y, then BA
aty, and AC="~*Y But (47. I. El.) BC?=BA? +AC?, which being expressed algebraically, becomes 62 = atymo + a-2ax+*+-+-y?
Likewise, from a known property of right angled triangles, BCXAD=BAXAC; that
is bx= (******=)
a-2axtx?? This last equation being multiplied by 2, and added to the former, gives b2+2bx =a?—2ax+x?, which being resolved according to the rules of Part I. Chap. 5. gives x=a+b72ab +262.
To construct this: a+b is the sum of the perimeter and perpendicular, and is given;
✓ 2ab +2b2=vat6x 2b is a mean proportional between a +b and 2b, and may be found ; therefore, from the sum of the perimeter and perpendicular, subtract the mean proportional between the said fum and double the base, and the remainder will be the perpendicular required.
From the base and perpendicular, the right angled triangle is easily constructed.
In numbers, let BA +AC +AD=18.8 =a; BC= 10 =b; then AD = a +bV Zab+262 = 28.8—V576=4.8=x, and BA+AC=14. By either of the first
equations, ya=262 +2ax—a—x2=4 and
y= BA — AC = 2; therefore BA = 8, and AC=6.
The geometrical expression of the roots of final equations arising from problems, may be found without resolving them, by the intersection of geometrical lines. Thus, the roots of a quadratic are found by the intersections of the circle and straight line, those of a cubic and biquadratic, by the intersection of two conic sections, &c.
The solution of problems may be effected also by the intersections of the loci of two intermediate equations without deducing a final equation : But these two last methods can only be understood by the doctrine of the loci of equations.
INES which can be mathematically
treated of, must be produced according to an uniform rule, which determines the position of every point of them.
This rule constitutes the definition of any line from which all its other properties are to be derived.
A straight line has been considered as so simple, as to be incapable of definition. The curve lines here treated of, are supposed to be in a plane, and are defined either from the section of a solid by a plane, or more universally by some continued motion in a plane, according to particular rules. Any of the properties which are shewn to belong peculiarly to such a line, may be assumed also as the definition of it, from which all the others, and even what, upon other occasions may have been considered as the primary definition, may be demonstrated. Hence lines may be defined in various methods, of which the most convenient is to be determined by the purpose in view. The simplicity of a definition, and the ease with which the other properties can be derived from it, generally give a prefe