coordinate geometry

[denian]

just wanna ask a simple question

a quadrilateral has the vertices A ( 1,4 ), B (9,5 ) , C ( 5,-2) and D (-3,-3). show that the quadrilateral is a rhombus

what i do is find gradient AB, CD, AD and BC.
and then state which and which have the same gradient... and hence the opposite sides are parallel.

is that all i should do?

[AKG]

Unless they're doing something tricky, the sides should be AB, BC, DC, and AD. Show that AB = DC, BC = AD, and |AB| = |BC|, and you're done. Basically a rhombus as two defining properties:

1) all four sides have the same length
2) opposite sides are parallel

In truth, it's sufficient to simply show that all four sides have the same length, and property 2 will follow, so that's one alternative approach. I would say try both, and see which one feels easier/more efficient. As for your approach, I don't know what you mean by gradient, so I can't say whether it will work or not.

[maverick280857]

The two sufficient conditions to prove are:

1. All 4 sides are equal (opposite sides are parallel).
2. The diagonals are not equal.

AKG: By gradient, denian means slope of a line joining (x_{1}, y_{1}) and (x_{2}, y_{2}) defined as:


m = \frac{y_{2}-y_{1}}{x_{2}-x_{1}}


Cheers
Vivek

[denian]

thank you!