6.8. Quadrilateral Classification http://www.ck12.org
Once you have a guess for what type of quadrilateral it is, your job is to prove your guess. To prove that a quadrilateral
is a parallelogram, rectangle, rhombus, square, kite or trapezoid, you must show that it meets the definition of that
shape OR that it has properties that only that shape has.
If it turns out that your guess was wrong because the shape does not fulfill the necessary properties, you can guess
again. If it appears to be no type of special quadrilateral then it is simply aquadrilateral.
The examples below will help you to see what this process might look like.
Example A
Determine what type of parallelogramT U NEis:T( 0 , 10 ),U( 4 , 2 ),N(− 2 ,− 1 ), andE(− 6 , 7 ).
Thislookslike a rectangle. Let’s see if the diagonals are equal. If they are, thenT U NEis a rectangle.
EU=
√
(− 6 − 4 )^2 +( 7 − 2 )^2 T N=
√
( 0 + 2 )^2 +( 10 + 1 )^2
=
√
(− 10 )^2 + 52 =
√
22 + 112
=
√
100 + 25 =
√
4 + 121
=
√
125 =
√
125
If the diagonals are also perpendicular, thenT U NEis a square.
Slope ofEU=−^76 −−^24 =− 105 =−^12 Slope ofT N=^100 −−((−− 21 ))=^112
The slope ofEU 6 =slope ofT N, soT U NEis a rectangle.
Example B
A quadrilateral is defined by the four linesy= 2 x+1,y=−x+5,y= 2 x−4, andy=−x−5. Is this quadrilateral
a parallelogram?
To check if its a parallelogram we have to check that it has two pairs of parallel sides. From the equations we can
see that the slopes of the lines are 2,−1, 2 and−1. Because two pairs of slopes match, this shape has two pairs of
parallel sides and is a parallelogram.