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Below are graphs of three different rhombi with their diagonals marked and the following vertices:

Rhombus 1: A (-6,2); B (1,3); C (6,-2); D (-1,-3)
Rhombus 2: A (-3,5); B (1,6); C (2,2); D (-2,1)
Rhombus 3: A (0,4); B (3,6); C (6,4); D (3,2)

On your own paper copy, and complete the table shown below finding the slopes of the diagonals of each rhombus.

	Slope of diagonal AC	Slope of diagonal BD
Rhombus 1
Rhombus 2
Rhombus 3

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	Slope of diagonal AC	Slope of diagonal BD
Rhombus 1	- 1 3	3
Rhombus 2	- 3 5	5 3
Rhombus 3	0	undefined

Based on the results of your data, what appears to always be true about the relationship between the slopes of the diagonals of a rhombus?

Rrecord your data in your notes.

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The slopes are negative reciprocals, which makes the diagonals always perpendicular to each other.

If you were given only the coordinates of the vertices, how could you determine if a quadrilateral was a rhombus? Record your data.

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Use the coordinates to find the slopes of the diagonals to see if they were negative reciprocals, then use the coordinates with the distance formula to find the length of the sides.

Based on this experiment, look at the other quadrilaterals in this flow chart below. First, notice how the quadrilaterals are grouped. There are three sections below quadrilateral: parallelogram, kite, and trapezoid. Each grouping is a special quadrilateral of the one above it. Under parallelogram are rectangle, rhombus, and square. Below kite is a right kite and below the trapezoid are two special trapezoids: isosceles and right trapezoids. Which do you think would have slopes that are perpendicular to each other?

Source: Properties of Quadrilaterals, Mr. Besworth, Mayfield Secondary School

	Diagonals Perpendicular? Always, Sometimes, Never	Check Your Answer
Quadrilateral		Interactive button. Assistance may be required. Check Sometimes
Parallelogram		Interactive button. Assistance may be required. Check Sometimes
Rectangle		Interactive button. Assistance may be required. Check Sometimes
Rhombus	Always
Square		Interactive button. Assistance may be required. Check Always
Kite		Interactive button. Assistance may be required. Check Always
Right Kite		Interactive button. Assistance may be required. Check Always
Trapezoid		Interactive button. Assistance may be required. Check Never
Isosceles Trapezoid		Interactive button. Assistance may be required. Check Never
Right Trapezoid		Interactive button. Assistance may be required. Check Never

Given the following coordinates of the vertices of a quadrilateral, determine algebraically if the figure could be a rhombus.

Quadrilateral	Vertex A	Vertex B	Vertex C	Vertex D	Rhombus? Possible/ Impossible	Check Your Answer
1	(2.5, 4)	(2, 2)	(4, 2.5)	(4, 4)		Interactive button. Assistance may be required. Check Impossible
2	(1.5, 4.5)	(2.5, 2)	(3, 5)	(5.5, 4)		Interactive button. Assistance may be required. Check Possible
3	(1.5, 4.5)	(1.5, 2.5)	(4, 2)	(3.5, 2)		Interactive button. Assistance may be required. Check Possible
4	(-1, 2)	(-2, 0)	(-1, -2)	(0, 0)		Interactive button. Assistance may be required. Check Impossible

What other element would you need to show the above "possible" rhombi are in fact rhombi?

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The lengths of the sides of the quadrilateral would all need to be the same.