Classify Polygons using coordinate geometry.
SM1 10-3
DATE____________
Polygons are shapes with three or more sides. They are classified as triangles, quadrilaterals, and
pentagons to name a few. When given the coordinates we can classify the polygons. Below are
some properties or requirements for classifying certain quadrilaterals and triangles.
Quadrilaterals
Property
Parallelogram
The diagonals
X
bisect each other
The diagonals
are congruent
The diagonal
bisects a pair of
opposite angles
The diagonals
are
perpendicular
Opposite sides
X
are congruent
Opposite sides
X
are parallel
All four sides
are congruent
All four angles
are 90 degrees
Triangles
Altitude
Rectangle
X
Square
X
X
X
X
X
X
X
X
X
X
X
X
x
X
X
X
X
A line that starts on a side of a
triangle and is perpendicular to
its, height
Two lines that Intersect at 90
degrees
Isosceles
Bisector
Cuts a line segment exactly in
half
Equilateral
Perpendicular
bisector
A line that starts on the midpoint
of a side of a triangle and
Intersects at 90 degrees
A line that starts at the vertex of a
triangle and goes to the exact
middle of the opposite side
Perpendicular
Median
Rhombus
X
Scalene
A triangle with
two congruent
sides
A triangle with
no sides that are
congruent
A triangle with
all three sides
equilateral
Even though it is not shown in the examples it is highly recommended that you graph the
points just to see what the shape looks like. A simple sketch gives you an idea of what it
looks like; but can in no way justify an answer.
EXAMPLE 1
Determine whether parallelogram ABCD is a rectangle A(-6, 9), B(5, 10), C(6, -1),
D(-5, -2).
SOLUTION
The diagonals of a rectangle are congruent, so if we use the distance formula to find the
length of the diagonals, we can determine if it is a rectangle.
AC  (6  6)2  (9  (1))2
AC  (12)2  (10)2
AC  144 100
AC  244  15.62
Since ABCD is a parallelogram and the diagonals are congruent, ABCD is a rectangle.
Once you know about parallel and perpendicular slopes, you can use this knowledge to prove
coordinate points are rectangles or parallelograms. Parallelograms have opposite sides that are
parallel. Rectangles have opposite sides that are parallel and the slopes must be perpendicular
for adjacent sides.
EXAMPLE 2
The coordinates of the vertices of RSTV are R(1, 1), S(3, 6), T(8, 8), V(6, 3). Determine
If RSTV is a parallelogram. Is it a rectangle as well?
SOLUTION
The opposite sides of a parallelogram are parallel. We can determine if RSTV is a
parallelogram by comparing the slopes of the sides.
uur 6  1 5
uur 8  6 2
RS 
or
ST 
or
31 2
83 5
Slope of
Slope of
uur 3  8 5
TV 
or
68 2
Slope of
uuur 3  1 2
RV 
or
6 1 5
Slope of
Since the opposite sides have the same slope, RV || TV and ST || RV . Therefore, RSTV is
a parallelogram. This is not a rectangle though since the slopes are not inverse
reciprocals.
EXAMPLE 3
ABC has vertices A(-3 ,10) B(9, 2) and C(9, 15). Classify the triangle.
SOLUTION
In example 4 from lesson 10-2, we found the lengths of the sides of a triangle. They are
AB  (3  9)2  (10  2)2  12 2  8 2  144  64  208
BC  (9  9)2  (15  2)2  0  132  13
CA  (9  (3))2  (15  10)2  12 2  5 2  144  25  169  13
This triangle is an isosceles triangle since two of the sides are congruent.