The document discusses classifying quadrilaterals using their coordinate graphs. It explains that to identify special types of quadrilaterals, one can use the distance and slope formulas to check if sides are parallel or congruent. An example problem classifies a quadrilateral as a square by showing its four right angles using slopes and that all four sides have equal lengths using the distance formula.

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

2. 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 sidesRhombus: Four congruent sides
Square: Rectangle and rhombus
Trapezoid: One pair of parallel sides
Kite: Two pairs of consecutive congruent
sides

3. To show sides are congruent, use the
distance formula:
To show sides are parallel, use the slope
formula:
( ) ( )
2 2
2 1 2 1d x x y y= − + −
2 1y y
m
−
=
−
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 1
m
x x
=
−

4. 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)?

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:
( )
( )TO
2 2 4
m
3 6 3
− −
= =
− − −
( )OY
1 2 3
m
1 3 4
− −
= = −
− −
( )5 1 4 4− − − −
opposite
reciprocals → 90°
→parallellines
( )
YS
5 1 4 4
m
2 1 3 3
− − − −
= = =
− − −
( )
( )ST
2 5 3
m
6 2 4
− − −
= = −
− − −
opposite
reciprocals → 90°
equalslopes→

6. We now know this is a rectangle. 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:
2 2
TO 3 4 5= + =
( )
22
OY 4 3 5= + − =
Since it has four right angles and four
congruent sides, TOYS is a square.
( ) ( )
2 2
YS 3 4 5= − + − =
( )
2 2
ST 4 3 5= − + =