In parallelogram ABCD, A (0,0),B(a,b)and D(c,0)are three of it's vertices, what are the coordinates of C in terms of a,b,c

To find the coordinates of point C in parallelogram ABCD, we need to use the properties of a parallelogram.

In a parallelogram, opposite sides are parallel, which means that AB is parallel to CD and AD is parallel to BC.

To find point C, we can use the fact that AB is parallel to CD. We know that A (0,0) and B (a,b) are two points on the line AB. The distance between these two points is given by:

AB = √((a - 0)^2 + (b - 0)^2)

Similarly, since C (x, y) and D (c, 0) are two points on the line CD, the distance between them is:

CD = √((x - c)^2 + (y - 0)^2)

Since AB is parallel to CD, the distances AB and CD are equal:

AB = CD

Substituting the values, we have:

√((a - 0)^2 + (b - 0)^2) = √((x - c)^2 + (y - 0)^2)

Squaring both sides of the equation, we get:

(a - 0)^2 + (b - 0)^2 = (x - c)^2 + (y - 0)^2

Simplifying further, we get:

a^2 + b^2 = x^2 - 2cx + c^2 + y^2

Since we want to find the coordinates of point C, we need to find x and y in terms of a, b, and c.

By comparing the coefficients of a^2, b^2, and c^2 on both sides of the equation, we get:

x^2 - 2cx = 0
x(x - 2c) = 0

Since the x coordinate of point C is x = 0 or x = 2c, if x = 0, then the y coordinate of point C will be:

a^2 + b^2 = 0^2 + y^2
y^2 = a^2 + b^2
y = ± √(a^2 + b^2)

Therefore, the coordinates of point C are:

C(0, ± √(a^2 + b^2)) or
C(2c, ± √(a^2 + b^2))

A(o,o),B(a,b),C(b,b),D(c,0).