Coordinate Plane Quads

The student will be able to (I can):

• Recognize special quadrilaterals from their graphs

• Use the distance and slope formulas to show that a quadrilateral is a special quadrilateral

If we are just given the coordinates of a quadrilateral, or even from the graph, it can be tricky to classify it. It’s usually easiest to go back to the definitions:

Parallelogram: Two pairs of parallel sides

Rectangle: Four right angles

Rhombus: Four congruent sides

Square: Rectangle and rhombus

Trapezoid: One pair of parallel sides

Kite: Two pairs of consecutive congruent sides

To show sides are congruent, use the distance formula:

To show sides are parallel, use the slope formula:

Hint: You might notice that both formulas use the differences in the x and y coordinates. Once you have figured the differences for one formula, you can just use the same numbers in the other formula.

( ) ( )2 2

2 1 2 1d x x y y= − + −

2 1

2 1

y ym

x x

−=

−

Example: What is the most specific name for the quadrilateral formed by T(—6, —2), O(—3, 2), Y(1, —1), and S(—2, —5)?

We might suspect this is a square, but we still have to show this. To show that it is a rectangle, we look at all of the slopes:

Two sets of equal slopes prove this is a parallelogram. Four 90° angles prove this is a rectangle.

( )

( )TO

2 2 4m

3 6 3

− −= =− − −

( )OY

1 2 3m

1 3 4

− −= = −

− −

( )YS

5 1 4 4m

2 1 3 3

− − − −= = =

− − −

( )

( )ST

2 5 3m

6 2 4

− − −= = −− − −

opposite reciprocals → 90°

opposite reciprocals → 90°

equa

l slope

s →

para

llel lines

To prove it is a square, we also have to show that all the sides are congruent. Since we have already set up the slopes, this will be pretty straightforward:

Since it has four right angles and four congruent sides, TOYS is a square.

2 2TO 3 4 5= + =

( )22OY 4 3 5= + − =

( ) ( )2 2

YS 3 4 5= − + − =

( )2 2ST 4 3 5= − + =