Recognize special quadrilaterals from their graphs Use the distance and slope formulas to show whether a quadrilateral is a special quadrilateral

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.
( )
( )
2 2 4
3 6 3TO
m
− −
= =
− − −
( )
1 2 3
1 3 4OY
m
− −
= = −
− −
( )5 1 4 4
2 1 3 3YS
m
− − − −
= = =
− − −
( )
( )
2 5 3
6 2 4ST
m
− − −
= = −
− − −
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
3 4 5TO = + =
( )
22
4 3 5OY = + − =
( ) ( )
2 2
3 4 5YS = − + − =
( )
2 2
4 3 5ST = − + =