:J4 i/. ) '//' OSOMKTin' [di. n 



6. Prove that the points (0, (K), l\ *\ and (:'.. *\ form an equi- 

 lateral triangle. 



7 < Mie end of a line whose length i- n is at the point (~4, 8), the 

 ordinal** of the other end is 3; what is its abscissa? 



8 Kxpreas by an equation the fact that the point P=(x, y) is at the 

 distance 3 from the point ( .'. :;> ; from the point (0,0). 



9. Kxpreas by an equation the fact that the point P=(x, y) is 

 equidistant from the point* ( -J, 3) and (7,5). 



10. Kind the slopes of the lines which join the following pairs of 

 points: (3, 8) and ( 1, 1 ) ; ( J, -3) and (7, 9); (1, -4) and (~3, 6) ; (4, ) 

 and (-2, 1). 



2B. One great advantage of the analytic method of solv- 

 ing problems lies in the fact that the analytic results which 

 are obtained from the simplest arrangement of the geomct ri<- 

 figure with reference to the coordinate axes are, from tin- 

 very nature of the method, equally true for all other arrange- 

 ments. Thus formulas [1], [2], and [3] can be most readily 

 obtained if the points are all taken in quadrant I, i.e., \\itli 

 their coordinates all positive ; but because of the convention 

 adopted concerning the signs as essential parts of the coordi- 

 nates, these formulas remain true for all possible positions of 

 PI and Pj. By drawing the figures and making the proofs 

 when PI and P, are taken in various other positions, tin? 

 student should assure himself of the generality of formulas 

 f 1 ]. [_'], and [3] of articles 26 and 27. 



29. The area of a triangle. 



1. Rectangular coordinates. Given a triangle with the, 

 vertices P l = (* y,), P, '= (x* y,), and P, = (3* y,) ; to find 

 its area in terms of 2^, x* s,, y,, y* and y,. Draw the onli- 

 nates 3f,P,, M t P* and 3/,P,, in the second figure extend 

 3f,P, and Mf % to meet a line through P 2 parallel to the 

 x-axis. If A represents the area of the triangle in the first 

 figure, then : 



