Question

The coordinates of parallelogram PVWZ are P(0, 0), V(a, b), and Z(c, 0). Find the coordinates of W without using any new variables.

Answer 1

Co-ordinates of W must be (c+a,b).

Explanation:

We are given that,

PVWZ is a parallelogram with vertices P(0,0), V(a,b) and Z(c,0).

Since, PVWZ is parallelogram.

The possible line segments are PV, VW, WZ and ZP.

Also, as the opposite sides must be parallel for the parallelogram, the co-ordinates will move accordingly.

So, we see from the figure that,

Co-ordinates of W must be (c+a,b).

Explanation:

Answer 2

We can call the difference of two points a vector.

A parallelogram has the property that the vectors of opposite sides are the same. Identical vectors is one way of thinking about what parallel and congruent sides means.

So in PVWZ we have opposite sides PV and ZW and also another pair of opposite sides VW and PZ.

Note we have to be a little careful about the order of the vertices. We want the vectors to be equal, which means they have to point in the same direction. If we have chosen PV and WZ the resulting vectors would be negative of each other.

In vectors

PV=ZW

Since a vector is a difference of points,

V - P = W - Z

W = V - P + Z = V + Z

because P(0,0) doesn't contribute anything to the sum.

W = V + Z = (a,b) + (c,0) = (a+c, b)

Answer: (a+c,b)

Check:

We can check the vectors from the other pair of parallel sides:

VW=PZ

W - V = Z - P

W = V + Z

as before.