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Answers for 6-2C

Geometry Section 6-2D Quadrilaterals and Coordinate proofs Page 433 Be ready to grade 6-2C Quiz Thursday. Answers for 6-2C. Rectangle Rhombus Parallelogram Square x = 30, y = 60 (2 pts.) x = 8, y = 22 (2 pts.) x = 8 (1 pt.) Not possible Drawing – a rectangle No – (2 pts.)

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Answers for 6-2C

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  1. Geometry Section 6-2D Quadrilaterals and Coordinate proofsPage 433Be ready to grade 6-2CQuiz Thursday

  2. Answers for 6-2C • Rectangle • Rhombus • Parallelogram • Square • x = 30, y = 60 (2 pts.) • x = 8, y = 22 (2 pts.) • x = 8 (1 pt.) • Not possible • Drawing – a rectangle • No – (2 pts.) • Diagonals bisect each other • Diagonals bisect and are perpendicular • Sketch with no lines of symmetry • Sketch with 2 lines of symmetry • Sketch with 4 lines of symmetry 18 pts. possible

  3. GRADE SCALE – 18 POSSIBLE 17.5 – 97% 17 – 94% 16.5 – 92% 16 – 89% 15.5 – 86% 15 – 83% 14.5 – 81% 14 – 78% 13.5 – 75% 13 – 72% 12.5 – 69% 12 – 67% 3

  4. Coordinate Geometry: Many proof can be made easier using coordinate geometry. To use this method, we first place the figure on a coordinate plane so that one vertex is at the origin and one side is on an axis. Pg.433

  5. Coordinate Geometry: Review: How can you calculate the slope of a line on a coordinate plane? rise run Slope = Pg.433 y2 –y1 x2 – x1

  6. Coordinate Geometry: Review: What is true about the slopes of perpendicular lines? The slopes of the 2 lines will be negative reciprocals of each other. Pg.433 If we want to prove that sides of a quadrilateral are perpendicular, we will prove their slopes multiplied equal -1.

  7. Coordinate Geometry: Review: What is true about the slopes of parallel lines? The slopes of the 2 lines will be identical. Pg.433 If we want to prove that sides of a quadrilateral are parallel, we will prove their slopes are the same.

  8. Coordinate Geometry: If I know that one point of a shape will always be on (0,0) then we can use variables to indicate the coordinates of the other points. If E is at (0,0) and we know that F is 9 units away, what are the coordinates of F? Pg.433 E F (9,0)

  9. Coordinate Geometry: If I know that one point of a shape will always be on (0,0) then we can use variables to indicate the coordinates of the other points. If E is at (0,0) and we know that F is 18 units away, what are the coordinates of F? Pg.433 E F (18,0)

  10. Coordinate Geometry: If I know that one point of a shape will always be on (0,0) then we can use variables to indicate the coordinates of the other points. If E is at (0,0) and we know that F is x units away, what are the coordinates of F? Pg.433 F E (x,0)

  11. Coordinate Geometry: If I know that one point of a shape will always be on (0,0) then we can use variables to indicate the coordinates of the other points. If E is at (0,0) and we know that F is a units away, what are the coordinates of F? Pg.433 E F (a,0)

  12. Coordinate Geometry: If I know that one point of a shape will always be on (0,0) then we can use variables to indicate the coordinates of the other points. If E is at (0,0) and we know that G is 9 units away, what are the coordinates of G? G Pg.433 E (0,9)

  13. Coordinate Geometry: If I know that one point of a shape will always be on (0,0) then we can use variables to indicate the coordinates of the other points. If A is at (0,0) and we know that B is b units away, what are the coordinates of B? B Pg.433 A (0,b)

  14. Coordinate Geometry: If I know that one point of a shape will always be on (0,0) then we can use variables to indicate the coordinates of the other points. If I wanted to move B 3 places to the right, what would it’s coordinate be? B Pg.433 A (0+3,b) (3,b)

  15. Coordinate Geometry: If I know that one point of a shape will always be on (0,0) then we can use variables to indicate the coordinates of the other points. If I wanted to move B 4 places to the left, what would it’s coordinate be? B Pg.433 A (0-4,b) (-4,b)

  16. Coordinate Geometry: If I know that one point of a shape will always be on (0,0) then we can use variables to indicate the coordinates of the other points. If I wanted to move B x places to the left, what would it’s coordinate be? B Pg.433 A (0-x,b) (-x,b)

  17. Coordinate Geometry: If I know that one point of a shape will always be on (0,0) then we can use variables to indicate the coordinates of the other points. If we know no specific numbers except the (0,0), we use variables and equations to give the other coordinates. G Pg.433 E F We’ll work across the bottom first. E = (0,0) F = (a,0) G = (b,c)

  18. Coordinate Geometry: To find point D, we take the labeled height and an equation to show the shift left or right. G H Pg.433 E = (0,0) E F F = (a,0) G = (b,c) H = (a+b,c)

  19. Example: Place a parallelogram on the coordinate plane. Whenever possible, place vertices on the axes. Let two vertices be the origin and (a,0). Pg.433 Place a vertex at (b,c). The last vertex insures that opposite sides have the same slope. Choose (a+b,c) to make another horizontal side and a second side of slope c/b.

  20. Explore: To use a coordinate proof, we will not use the 2 column proof. We will use the paragraph form. Place square ABCD, with side length a on a coordinate plane. Label the vertices and give their coordinates. Pg.435 A –B –C –D –

  21. Explore: To use a coordinate proof, we will not use the 2 column proof. We will use the paragraph form. Use the slope formula to find the slopes of the diagonals AC and BD. Pg.435

  22. Explore: To use a coordinate proof, we will not use the 2 column proof. We will use the paragraph form. Use your results from the previous screen to show that AC and BD are perpendicular. Explain. Pg.435

  23. Reflect: What properties can be used to show that lines are parallel or perpendicular when doing a coordinate proof? Pg.435 If they are parallel, slopes will be equal. If they are perpendicular, slopes will be negative reciprocals.

  24. Reflect: Why is the use of coordinates a helpful strategy in some proofs? Pg.435 Coordinates can be used to calculate lengths and slopes. We can use that information to prove congruence, parallel and perpendicular.

  25. Reflect: In trapezoid MNPQ, you could give point P the coordinates (c,d). However, there is a better choice for them. What are they? Explain. Pg.435 (a,c) is better because the distance on the x axis is the same for pts. N and P. Hint: don’t choose a new variable if there’s any way to use the other variables. Sometimes, the problem will tell you not to introduce a new variable. In that case, make an equation of some kind.

  26. Exercises: The figure shown is an isosceles trapezoid. What are the coordinates of vertex C? (Hint: You will have to introduce one new variable.) What are the coordinates of vertex D? #1 Pg.435 C is (c,b) D is (c+a,0)

  27. Exercises: Assign coordinates to the vertices. Do not introduce any new variables. ABCD is a rectangle. #2 Pg.436 (b,a)

  28. Exercises: Assign coordinates to the vertices. Do not introduce any new variables. DEFG is a parallelogram. #3 Pg.436 (a+c,b)

  29. Exercises: Assign coordinates to the vertices. Do not introduce any new variables. DHIJ is equilateral. #4 Pg.436 (a, aÖ3)

  30. Exercises: Use the figure to verify each statement. The opposite sides of FGHJ are parallel. #5 Pg.436 Slope of HJ = Slope of GF = Slope of JF = Slope of HG =

  31. Exercises: Use the figure to verify each statement. The opposite sides of FGHJ are congruent. #6 Pg.436 JF = HG = HJ = GF =

  32. Exercises: Use a coordinate proof to prove the following:If a quadrilateral is a square, then its diagonals are congruent. The distance from (0,0) to (a,a) is Öa2+a2 = aÖ2 #8 Pg.436 The distance from (0,a) to (a,0) is Öa2+(-a)2 = aÖ2 The diagonals are congruent.

  33. Exercises: Use a coordinate proof to prove the following:If a quadrilateral is a parallelogram, then its diagonals bisect each other. The midpoint of DB is #9 Pg.436 The midpoint of AC is The midpoints are identical, so the diagonals bisect.

  34. Homework: Practice 6-2D Quiz Thursday

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