1. Unit 6: Connecting Algebra and Geometry through Coordinates Proving Coordinates of Rectangles and Squares

2. Characteristics of Rectangles and Squares (both are parallelograms) Rectangles: Opposite sides are parallel and congruent Opposite angles are congruent and consecutive angles are supplementary All four angles are right angles (90°) Diagonals bisect each other and are congruent Squares: All sides are congruent Opposite sides are parallel All four angles are right angles Diagonals bisect each other and are congruent Diagonals are perpendicular Diagonals bisect opposite angles

3. Using the Distance Formula We will be using the distance formula to prove that given coordinates form a square or a rectangle. Remember the distance formula is derived from the Pythagorean Theorem: Also recall that: Parallel lines have the same slope Perpendicular lines have slopes that are negative (opposite sign) reciprocals whose product is -1.

4. Example 1: Prove that the following vertices represent the vertices of a rectangle. P (3, 1) Q (3, -3) R (-2, -3) S (-2, 1) Practice proving that vertices represent particular geometric figures by using all possible characteristics. Use the chart for quadrilaterals to help you remember the properties. S P Q R

5. Example 1: Prove that the following vertices represent the vertices of a rectangle. P (3, 1) Q (3, -3) R (-2, -3) S (-2, 1) S P Q R

6. Example 1: Prove that the following vertices represent the vertices of a rectangle. P (3, 1) Q (3, -3) R (-2, -3) S (-2, 1) S P Q R

7. Example 1: Prove that the following vertices represent the vertices of a rectangle. P (3, 1) Q (3, -3) R (-2, -3) S (-2, 1) S P Q R

8. Example 1: Prove that the following vertices represent the vertices of a rectangle. P (3, 1) Q (3, -3) R (-2, -3) S (-2, 1) S P Q R

9. Example 1: Prove that the following vertices represent the vertices of a rectangle. P (3, 1) Q (3, -3) R (-2, -3) S (-2, 1) S P Q R

10. Example 1: Prove that the following vertices represent the vertices of a rectangle. P (3, 1) Q (3, -3) R (-2, -3) S (-2, 1) S P Q R

11. Example 1: Prove that the following vertices represent the vertices of a rectangle. P (3, 1) Q (3, -3) R (-2, -3) S (-2, 1)

12. Example 2: Prove that the following vertices represent the vertices of a square. P (5, 2) Q (2, 5) R (-1, 2) S (2, -1) S P Q R Practice proving that vertices represent particular geometric figures by using all possible characteristics. Use the chart for quadrilaterals to help you remember the properties.

13. Example 2: Prove that the following vertices represent the vertices of a square. P (5, 2) Q (2, 5) R (-1, 2) S (2, -1) S P Q R

14. Example 2: Prove that the following vertices represent the vertices of a square. P (5, 2) Q (2, 5) R (-1, 2) S (2, -1) S P Q R

15. Example 2: Prove that the following vertices represent the vertices of a square. P (5, 2) Q (2, 5) R (-1, 2) S (2, -1) S P Q R

16. Example 2: Prove that the following vertices represent the vertices of a square. P (5, 2) Q (2, 5) R (-1, 2) S (2, -1) S P Q R

17. Example 2: Prove that the following vertices represent the vertices of a square. P (5, 2) Q (2, 5) R (-1, 2) S (2, -1) S P Q R

18. Example 2: Prove that the following vertices represent the vertices of a square. P (5, 2) Q (2, 5) R (-1, 2) S (2, -1) S P Q R

19. Example 2: Prove that the following vertices represent the vertices of a square. P (5, 2) Q (2, 5) R (-1, 2) S (2, -1) S P Q R

20. Example 2: Prove that the following vertices represent the vertices of a square. P (5, 2) Q (2, 5) R (-1, 2) S (2, -1) S P Q R

21. Example 2: Prove that the following vertices represent the vertices of a square. P (5, 2) Q (2, 5) R (-1, 2) S (2, -1) S P Q R

22. Example 2: Prove that the following vertices represent the vertices of a square. P (5, 2) Q (2, 5) R (-1, 2) S (2, -1)

23. Example 3: Do the given vertices represent those of a rectangle? Why or why not? P (5, 2) Q (1, 9) R (−3, 2) S (1, −5) S P Q R What would be the most obvious why to begin if you do not have a diagram?

24. Example 3: Do the given vertices represent those of a rectangle? Why or why not? P (5, 2) Q (1, 9) R (−3, 2) S (1, −5) S P Q R

25. Summary of the Proof Process When you are told that vertices are those of a certain quadrilateral, you may assume that the properties of that quadrilateral are present. When you are simply told vertices, often you must determine if those vertices represent a specific type of quadrilateral. Begin with an easy property to rule out possible types, such as length. Proceed with each additional and required property to verify a type of quadrilateral. For all HW, you must first state which property you are testing and show all work to support your conclusions. If you have a graph, you may count vertical or horizontal units to determine length, otherwise you must use the distance formula. If you have vertical or horizontal segments, you may write undefined or 0 for the slope. Otherwise, you must use the slope formula. Clearly state your conclusions in a complete sentence.