6.8. Quadrilateral Classification http://www.ck12.org
- To be a rectangle a shape must have four right angles. This means that the sides must be perpendicular to each
other. From the given equations we see that the slopes are 2,−2, 2 and−2. Because the slopes are not opposite
reciprocals of each other, the sides are not perpendicular, and the shape is not a rectangle. - First, graphABCD. This will make it easier to figure out what type of quadrilateral it is. From the graph, we can
tell this is not a parallelogram. Find the slopes ofBCandADto see if they are parallel.
Slope ofBC=^5 − 1 (−− 41 )=−^63 =− 2
Slope ofAD=^3 −− 3 (−−^51 )=−^84 =− 2
We now knowBC||AD. This is a trapezoid. To determine if it is an isosceles trapezoid, findABandCD.
AB=
√
(− 3 − 1 )^2 +( 3 − 5 )^2 ST=
√
( 4 − 1 )^2 +(− 1 −(− 5 ))^2
=
√
(− 4 )^2 +(− 2 )^2 =
√
32 + 42
=
√
20 = 2
√
5 =
√
25 = 5
AB 6 =CD, therefore this is only a trapezoid.
- We will not graph this example. Let’s find the length of all four sides.
EF=
√
( 5 − 11 )^2 +(− 1 −(− 3 ))^2 F G=
√
( 11 − 5 )^2 +(− 3 −(− 5 ))^2
=
√
(− 6 )^2 + 22 =
√
62 + 22
=
√
40 = 2
√
10 =
√
40 = 2
√
10
GH=
√
( 5 −(− 1 ))^2 +(− 5 −(− 3 ))^2 HE=
√
(− 1 − 5 )^2 +(− 3 −(− 1 ))^2
=
√
62 +(− 2 )^2 =
√
(− 6 )^2 +(− 2 )^2
=
√
40 = 2
√
10 =
√
40 = 2
√
10
All four sides are equal. That means, this quadrilateral is either a rhombus or a square. The difference between the
two is that a square has four 90◦angles and congruent diagonals. Let’s find the length of the diagonals.
EG=
√
( 5 − 5 )^2 +(− 1 −(− 5 ))^2 F H=
√
( 11 −(− 1 ))^2 +(− 3 −(− 3 ))^2
=
√
02 + 42 =
√
122 + 02
=
√
16 = 4 =
√
144 = 12
The diagonals are not congruent, soEF GHis a rhombus.
Practice
- If a quadrilateral has exactly one pair of parallel sides, what type of quadrilateral is it?
- If a quadrilateral has two pairs of parallel sides and one right angle, what type of quadrilateral is it?