1. Coordinate Proofs
Lesson 6-2

2. Warm-up Are the following lines perpendicular, parallel, or neither?
y = 2x y = 2x x + y = x + 6y = 12
y = 3x x + 2y = 8 9x + 3y = y = -2x + 4

3. Refresher Are the following lines perpendicular, parallel, or neither?
(2, 4) and (3, 6) 3. (5, 7) and (2, 4) (0, 0) and (2, 1) (3, 5) and (1, 7)
(4, 1) and (3, 5) 4. (3, 5) and (8, 5) (1, 2) and (5, 3) (2, 7) and (2, -4)

4. Midpoint Formula
To calculate the midpoint between two points, you average the x’s to get the x-value and average the y’s to get the y-value.
The formula is
Example: Find the midpoint of (5, 3) and (1, -5).

5. Distance Formula
To calculate the distance between two points, we treat the distance between them as the hypotenuse of a right triangle. The difference in x’s is one side of the triangle and the difference in y’s is the other side. The formula is
Example: Find the distance between (3, 8) and (2, 4).

6. Let’s Practice What is the distance from (2, 4) to (5, - 3)?
What is the midpoint of (2, -1) and (-4, 3)
The distance from point (1, 1) to another point (x, 5) is 5 units. What is x?
The midpoint of (3, -2) and another point (x, y) is (-2, 5). What is the point (x, y)?

7. Classifying Triangles
Scalene: No sides have the same measure
Isosceles: Two sides have the same measure
Equilateral: Three sides have the same measure

8. Coordinate Proofs Tell if a triangle with vertices at
(-3,2), (-2,-2), and (1,-2)
is a scalene, isosceles, or equilateral triangle.

9. First Step: Plot the points and draw the triangle.
B: (-2,-2)
C: (1, -2) c b

10. For side AB use (-3,2) and (-2,-2)
Step Two: Use the distance formula to find the measures of the three sides
For side AB use (-3,2) and (-2,-2)
√( )2 + (2 - -2)2
√(-1)2 + (4)2
√1 + 16
√17
4.1

11. For side BC use (-2,-2) and (1,-2)
Step Two: Use the distance formula to find the measures of the three sides
For side BC use (-2,-2) and (1,-2)
√(-2 - 1)2 + ( )2
√(-3)2 + (0)2
√9 + 0 √9 3

12. For side AC use (-3,2) and (1,-2)
Step Two: Use the distance formula to find the measures of the three sides
For side AC use (-3,2) and (1,-2)
√(-3 - 1)2 + (2 - -2)2
√(-4)2 + (4)2 √
√32
5.7

13. Step Three Use your information
AB = 4.1
BC = 3
AC = 5.7
No side measures the same
The triangle is:
Scalene

14. Coordinate Proofs Tell if a triangle with vertices at
(-5,-3), (7,-3), and (1,5)
is a scalene, isosceles, or equilateral triangle.

15. First Step: Plot the points and draw the triangle.
b: (-5,-3)
c: (7, -3) b c

16. For side AB use (1,5) and (-5,-3)
Step Two: Use the distance formula to find the measures of the three sides
For side AB use (1,5) and (-5,-3)
√(1 - -5)2 + (5 - -3)2
√(6)2 + (8)2 √
√100 10

17. For side BC use (-5,-3) and (7,-3)
Step Two: Use the distance formula to find the measures of the three sides
For side BC use (-5,-3) and (7,-3)
√(-5 - 7)2 + ( )2
√(-12)2 + (0)2 √
√144 12

18. For side AC use (1,5) and (7,-3)
Step Two: Use the distance formula to find the measures of the three sides
For side AC use (1,5) and (7,-3)
√(1 - 7)2 + (5 - -3)2
√(6)2 + (8)2 √
√100 10

19. Step Three Use your information
AB = 10
BC = 12
AC = 10
Two sides measures the same
The triangle is:
Isosceles

20. Tell if a quadrilateral with vertices at
Coordinate Proofs
Tell if a quadrilateral with vertices at
A: (-1,4)
B: (-1,-2)
C: (5,-2)
D: (5,4)
is square

21. First Step: Plot the points and draw the triangle.
b: (-1,-2)
c: (5,-2)
d: (5,4) c b

22. For side AB use (-1,4) and (-1,-2)
Step Two: Use the distance formula to find the measures of the three sides
For side AB use (-1,4) and (-1,-2)
√( )2 + (4 - -2)2
√(0)2 + (6)2
√0 + 36
√36 6

23. For side BC use (-1,-2) and (5,-2)
Step Two: Use the distance formula to find the measures of the three sides
For side BC use (-1,-2) and (5,-2)
√(-1 - 5)2 + ( )2
√(-6)2 + (0)2
√36 + 0
√36 6

24. For side CD use (5,-2) and (5,4)
Step Two: Use the distance formula to find the measures of the three sides
For side CD use (5,-2) and (5,4)
√(5 - 5)2 + (-2 - 4)2
√(0)2 + (-6)2
√0 + 36
√36 6

25. For side AD use (-1,4) and (5,4)
Step Two: Use the distance formula to find the measures of the three sides
For side AD use (-1,4) and (5,4)
√(-1 - 5)2 + (4 - 4)2
√(-6)2 + (0)2
√36 + 0
√36 6

26. Step Three Use your information
AB = 6
BC = 6
CD = 6
AD = 6
All sides measures the same
The quadrilateral is:
A Rhombus
How would we prove it’s a square?