* Recognize special quadrilaterals from their graphs

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
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 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:
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 y
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:
Two sets of equal slopes prove this is a
parallelogram. Four 90° angles prove this
is a rectangle.
( )
( )TO
2 2 4
m
3 6 3
− −
= =
− − −
( )OY
1 2 3
m
1 3 4
− −
= = −
− −
( )
YS
5 1 4 4
m
2 1 3 3
− − − −
= = =
− − −
( )
( )ST
2 5 3
m
6 2 4
− − −
= = −
− − −
opposite
reciprocals → 90°
opposite
reciprocals → 90°
equalslopes→parallellines

6. 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 2
TO 3 4 5= + =
( )
22
OY 4 3 5= + − =
( ) ( )
2 2
YS 3 4 5= − + − =
( )
2 2
ST 4 3 5= − + =