. Plane and solid analytic geometry . \ ANALYTIC GEOMETRY % - -«/. Mid-Point of a Line-Segment. Let Pi, with the coordi-y- nates (xi, 2/1)5 and P2: (x2, y-y) be the extremities of a line-seg-ment. It is desired to find thef|-(^22/2) coordinates of the point P whichbisects P1P2. Let the coordinates of P be{x, y). It is evident that thedirected line-segment PiP isequal to the directed line-seg-ment PP2. Hence the projec- tion of P^P on the axis of a;, or X — oji, must equal the projection of PP2 on that axis, or ^2 — a?: *C ~~~ Jbt - •- •^O ^~~ w*. Fig. 8 Hence Xi 4- X9 Similar considerations ap
. Plane and solid analytic geometry . \ ANALYTIC GEOMETRY % - -«/. Mid-Point of a Line-Segment. Let Pi, with the coordi-y- nates (xi, 2/1)5 and P2: (x2, y-y) be the extremities of a line-seg-ment. It is desired to find thef|-(^22/2) coordinates of the point P whichbisects P1P2. Let the coordinates of P be{x, y). It is evident that thedirected line-segment PiP isequal to the directed line-seg-ment PP2. Hence the projec- tion of P^P on the axis of a;, or X — oji, must equal the projection of PP2 on that axis, or ^2 — a?: *C ~~~ Jbt - •- •^O ^~~ w*. Fig. 8 Hence Xi 4- X9 Similar considerations apply to the projections on the axisof y, and consequently 2 We have thus obtained the following result: The coordinates(cc, y) of the point P ichich bisects the line-segment P1P2 are givenby the equations: (1) X _^l+^2 ^ 2 EXERCISES 1. Determine the coordinates of the mid-point of each ofthe line-segments given by the pairs of points in Ex. 1, § figures and check your answers. ^ -3v. Find the mid-points of the sides of the triangle mentionedin Ex. 2, § 3, and check by a figure. 3. Determine the coordinates of the mid-point of the linejoining the points (a + b, a) and (a — b, b). 4. Show that the diagonals of the parallelogram of Ex. 4,§ 4 bisect each other. COORDINATES. CURVES AND EQUATIONS 17 6. Division of a Line-Segment in Any Ratio.* Let it be re-quired to Und the coordinates (x, y) of the point P whichdivides the line-segment PxP^ in an arbitrary ratio, mi/m2: t PP2 TH Obviously the projections of P^P and PP2 on the axis o
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