EXAMPLE 4 Use coordinate geometry

SOLUTION

One way is to show that a pair of sides are congruent and parallel. Then apply Theorem 8.9.First use the Distance Formula to show that AB and CD are congruent.

AB = =[2 – (–3)]2 + (5 – 3)2 29

CD = (5 – 0)2 + (2 – 0)2 = 29

Show that quadrilateral ABCD is a parallelogram.

EXAMPLE 4 Use coordinate geometry

Because AB = CD = 29 , AB CD.

Then use the slope formula to show that AB CD.

Slope of AB =5 – (3)

2 – (–3) = 25

Slope of CD = 2 – 05 – 0 =

25

Because AB and CD have the same slope, they are parallel.

AB and CD are congruent and parallel. So, ABCD is a parallelogram by Theorem 8.9.

ANSWER

EXAMPLE 4GUIDED PRACTICE for Example 4

6. Refer to the Concept Summary. Explain how other methods can be used to show that quadrilateral ABCD in Example 4 is a parallelogram.

SOLUTION

Find the Slopes of all 4 sides and show that each opposite sides always have the same slope and, therefore, are parallel.

Find the lengths of all 4 sides and show that the opposite sides are always the same length and, therefore, are congruent.

Find the point of intersection of the diagonals and show the diagonals bisect each other.