Calculus: Analytic Geometry and Calculus, with Vectors

1.3 Lines and linear equations 19

10 Any given rectangle can be placed upon the x, y coordinate system in such
a way that its vertices are(0,0), (0,a), (b,0), and (b,a). Prove that if the diag-
onals are perpendicular, then the rectangle is a square.
11 Sketch a figure showing the triangle having vertices at the points Pi(xl,y,),
P2(x2,y2), and Pi(xa,Y3) For each k = 1, 2, 3, mark the mid-point Qk of the side
opposite Pk and find the coordinates of Qk. Supposing that the line QsQ1 is not
vertical, calculate its slope and show that it is parallel to the line P1P2.
12 Prove analytically (by calculating slopes) that the mid-points of the sides
of a convex quadrilateral are vertices of a parallelogram. Remark: Taking ver-
tices at (xi,yl), (x2,y2), (xa,ya),(x4,y4) produces "symmetric" formulas.
13 Show that the lines having the equations

aix + b,y = cl

a2x + b2y = c2

are parallel if and only if a1b2 - a2b1 = 0. If the lines are not parallel, they
must intersect at a point P(x,y) whose coordinates satisfy both equations.
Assuming that the lines are not parallel, solve the equations to obtain the formulas

b2c1 - blc2

X a,b2 - a2b1'

alc2 - a2c1

alb2 - a2b1

for the coordinates of the point of intersection. Remark: Those who have for-

gotten how to solve systems of linear equations can recover by noticing that we

can multiply the first equation by b2 and the second by -b1 and then add the

results to eliminate y and obtain an equation that can be solved for x. This

process is known as the process of successive elimination.

14 Copy Figure 1.292 and then find the equations of the three medians of the

triangle and show that these medians intersect at the point (2h/3, 0). Remark:

Since the median placed upon the x axis could have been any median of the tri-

angle, this provides a proof that the three medians of a triangle intersect at the

point which trisects each of them.

15 Show that the lines obtained by giving constant values to kin the equation

2x + 3y + k = 0

are all parallel. Show that the line L having the equation

2(x-xl)+3(y-yi) =0

belongs to this family and contains the point (x1,yl)

16 Show that if the lines .4P and BP joining the points 11(1,2) and B(5,-4)

to P(x,y) are perpendicular, then

(x - 1) (x - 5) + (y + 4) (y - 2) =0.

Remark: Persons well acquainted with elementary geometry should know that

P must lie on the circle having the line segment .4B for a diameter.

17 Put the following equations into normal form and check the results by

drawing graphs showing the lines having the given equations and the line seg-

ments through the origin perpendicular to these lines.