Question

Given A(1, 1), B(0,5), C(4,4), and D(5, 0). Use the fact that if the diagonals of a parallelogram are perpendicular, then it is a rhombus to prove ABCD is a rhombus?

Answer 1

Please see below.

Explanation:

The two diagonals are `AC` and `BD`. As product of the slopes of two lines perpendicular to each other is `-1`, we intend to show that.

Now, slope of line joining two points `(x_1,y_1)` and `(x_2,y_2)` is given by `(y_2-y_1)/(x_2-x_1)` and therefore

Slope of `AC` is `(4-1)/(4-1)=3/3=1`

Slope of `BD` is `(0-5)/(5-0)=-5/5=-1`

as product of the slopes is `1xx(-1)=-1`, they form a rhombus.