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Unit 6: Connecting Algebra and Geometry through Coordinates

Unit 6: Connecting Algebra and Geometry through Coordinates. Proving Coordinates of Rectangles and Squares. Characteristics of Rectangles and Squares (both are parallelograms). Rectangles: Opposite sides are parallel and congruent

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Unit 6: Connecting Algebra and Geometry through Coordinates

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  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. P S 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) Property: Opposite sides are congruent. Find the length of each side: = 4 ; = 4 = 5 ; = 5 P S 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) Property: Opposite sides are parallel. Find the slopes of opposite sides: m = 0 ; m = 0 m= undefined; m = undefined P S 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) Property: Angles are 90ᵒ. Find the slope of adjacent sides: m = 0 ; m = undefined m= undefined; m = 0 P S 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) Property: Angles are 90ᵒ. Find the slope of adjacent sides: m = undefined ; m = 0 ; m= undefined; m = 0 P S 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) Property: Diagonals are congruent. Find the length of each diagonal: = = P S 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) Property: Diagonals bisect each other. Find the midpoints of each diagonal: Midpoint = = Midpoint = = Diagonals bisect each other (have the same mid-point). P S 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) PQRS is a rectangle because: and ; Opposite sides are congruent. and ; Opposite sides are parallel. , , , ; All angles are 90ᵒ. ; Diagonals are congruent. Midpoint = Midpoint ; Diagonals bisect each other.

  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) Practice proving that vertices represent particular geometric figures by using all possible characteristics. Use the chart for quadrilaterals to help you remember the properties. Q R P S

  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) Property: All sides are congruent. Find the length of each side: = = = = Q R P S

  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) Property: All sides are congruent. Find the length of each side: = = = = Q R P S

  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) Property: All sides are congruent Find the length of each side: = ; = ; = ; = Q R P S

  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) Property: Opposite sides are parallel. Find the slope of each side: m of = = = 1 m of = = = 1 Q R P S

  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) Property: Opposite sides are parallel. Find the slope of each side: m of = = = -1 m of = = = -1 Q R P S

  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) Property: All angles are 90ᵒ. Use the slopes of each side: m of = 1 ; m of = -1 m of = -1 ; m of = 1 m of = 1 ; m of = -1 m of = -1 ; m of = 1 Q R P S

  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) Property: Diagonals are congruent. Find the length of each diagonal: = 6 = 6 Q R P S

  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) Property: Diagonals bisect each other. Find the mid-point of each diagonal: Midpoint = = Midpoint = = Diagonals bisect each other (have the same mid-point). Q R P S

  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) Property: Diagonals are perpendicular. Find the slopes of each diagonal: m of = m of = Diagonals are perpendicular. Q R P S

  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) PQRS is a square because: ; All sides are congruent. and ; Opposite sides are parallel. , , , ; All angles are 90ᵒ. ; Diagonals are congruent. Midpoint = Midpoint ; Diagonals bisect each other. ; Diagonals are perpendicular.

  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) What would be the most obvious why to begin if you do not have a diagram? Q Check for congruency of opposite sides. = = = = Since all sides are congruent, this could be a rectangle (more specifically, a square). Now check slopes for 90ᵒ angles. R P S

  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) Q Check slopes for possible 90ᵒ angles. m of = = = m of = = = − Since adjacent angles are opposite signs only (and not reciprocals), these vertices do not represent those of a rectangle. What kind of quadrilateral is PQRS? R P S

  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.

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