Jeff K. answered • 05/26/20

Together, we build an iron base of understanding

Jasmina,

The question is unclear but I read the 4 points as: A (0, 6), B (4, 1), C (-4, 0), D (-8, 7).

Those 4 points, ABCD do form a convex quadrilateral.

But the 2 pairs of opposite sides are then not parallel, so it's not a rhombus!

Here's how to tackle. it.

(a) Find the slope of AB, using (y_{B} - y_{A})/(x_{B} - x_{A}). Do the same for CD. The 2 slopes should be equal, meaning AB || CD.

Do the same for AD and BC. They should also have the same slopes , so AD || BC.

Therefore, ABCD is a parallelogram.

To be a rhombus (a squashed over square), the 4 sides must also be equal in length.

Use the distance formula L = √((x_{1}^{2} - x_{2}^{2}) + (y_{1}^{2} - y_{2}^{2})) to calculate the lengths of the above pairs of sides. They should all be equal => ABCD is a rhombus.

(b) To show that the diagonals, AC and BD, are perpendicular, calculate their slopes. m_{AC} and m_{BD}.

If m_{AC} x m_{BD }= -1, then AC is ⊥ BD

Calculate m_{AC} and m_{BD }by (y_{C} - y_{A}) / (x_{C} - x_{A}) and (y_{B} - y_{A} / (x_{B} - x_{D})

Check that m_{AC} x m_{BD }= -1 which proves that AC ⊥ BD, as required.