Question

What are ways using coordinate geometry, that I could determine that this is a trapezoid?

Answer 1

One method to see it's a trapezoid is to find the slope of lines BC and AD.

The slope formula is

m = (y2-y1)/(x2-x1)

You should find that BC and AD both have the same slope (of -1), so that means the lines are parallel. That proves we have a trapezoid.

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To prove this trapezoid is isosceles, you can use the distance formula

$d = \sqrt{ \left(x_1-x_2\right)^2 + \left(y_1-y_2\right)^2}$

to find the lengths of AB and CD (the two non-parallel sides). You should find that AB = CD.

Because AB and CD are horizontal and vertical respectively, this means you can simply count out the spaces to find that AB and CD are 3 units each. For any other rotated version of this trapezoid, use the distance formula instead.