Formulas and the coordinate plane Distance formula

To determine whether sides are congruent. To determine if diagonals are congruent.

Midpoint formula To determine the coordinates of the midpoint of a side. To determine whether diagonals bisect each other.

Slope formula To determine whether opposite sides are parallel. To determine whether diagonals are perpendicular. To determine whether sides are perpendicular.

Is triangle ABC scalene, isosceles, or equilateral.

(0,1), (4,4), (7,0)A B andC2 2(4 0) (4 1)AB = − + −

16 9= + 25 5= =2 2(7 4) (0 4)BC = − + −

9 16= +2 2(0 7) (1 0)CA = − + − 49 1= +50= 5 2=

ABC is an isosceles triangle.

Is parallelogram ABCD a rhombus? Explain.

5 0 5

4 ( 2) 6slopeof AC

−= =− −

1 4 3

2 0 2slopeof BD

− −= =−

Since the slopes are not opposite reciprocals, the diagonals are not perpendicular. Therefore, ABCD is not a rhombus.

A kite is shown. What is the most precise classification of the quadrilateral formed by connecting the midpoints of the sides of the kite?

4 0 0 4, ( 2, 2)

2 2A

− + + = = − ÷

0 8 4 0, (4,2)

2 2B

+ + = = ÷ 8 0 0 ( 4)

, (4, 2)2 2

C+ + − = = − ÷ 0 ( 4) 4 0

, ( 2, 2)2 2

D+ − − + = = − − ÷

| 4 ( 2) | 6AB = − − =

| 2 2 | 4BC = − − =

| 2 4 | 6CD = − − =

| 2 ( 2) | 4DA = − − =

Opposite sides are congruent. Sides are vertical and horizontal, making them perpendicular. Therefore, ABCD is a rectangle.

PRACTICE PRACTICE

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