# Coordinate geometry

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

and then state which and which have the same gradient... and hence the opposite sides are parallel.

is that all i should do?

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AKG
Homework Helper
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.

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

thank you!