Exploring Coordinate Geometry: Points, Lines, and Equations Distances and Midpoints

1. Plot the following points: • A(0, 4) • B(-3, 0) • C(3, 5) • D(-8, -11) • E(7, -5) • F(-3.5, 7.5)

2. Coordinate Geometry Unit Essential Question: How can you use the coordinate plane to solve problems and demonstrate properties?

3. The Coordinate Plane Essential Question: How do you calculate distance, midpoint, and slope on a coordinate plane?

4. Origin • x-axis • y-axis • Quadrants • Coordinates • Positive • Negative

5. Distance Formula

6. Why?

7. Midpoint • What is the midpoint of 7 and 15? • What is the midpoint of (2, 7) and (4, 1)?

8. Midpoint…

9. Find the Midpoint:

10. Finding an Endpoint:

11. Distance? • Mary walks 3 miles north and 2 miles east. Her boyfriend John walks 2 miles south and 4 miles west. What is the straight line distance from their hearts?

13. Opener (Algebra Review)

14. Lines in the Plane Essential Question: How do the slopes of parallel and perpendicular lines relate?

15. Point-Slope form (writing eqs) • Line with slope, m = -2, through (3, -5)

16. Slope-Intercept Form • y = mx + b • m = • b =

17. Example

18. Example: • b(egin) = • m(ove)=

19. Equations: m = -1/2, (-4, -1) • Point-Slope • Slope Intercept

20. Standard Form: Ax + By = C

21. Example: • x-int (set y=0) • y-int (set x=0)

22. Transforming to Slope-Intercept • Solve for y: • b(egin) • m(ove)

23. Example: • b(egin) = • m(ove)=

24. Writing Equations through 2 pts • (10, 2) & (2, -2) • Step 1: Find Slope • Step 2: Combine pt & slope

25. Equation through: (7, -4), (-5, 2) • Slope: • Equation:

26. Horizontal & Vertical Lines • What is special? • Horizontal: • Vertical:

27. Horizontal & Vertical Lines

29. Graph the following Lines:

30. Slopes of Parallel & Perpendicular Lines Essential Question: How do the slopes of parallel and perpendicular lines relate?

31. Parallel Lines • Are these parallel?

32. Which is parallel? • Can you write the equation of a 3rd line that is parallel?

33. Writing equations of parallel lines

34. Perpendicular Lines • They are OPPOSITE RECIPROCAL slopes.

35. Are they Perpendicular?

36. Writing Perpendicular Equations • Step 1: Perpendicular Slope: • Step 2: Combine Point & Slope:

37. Perpendicular Lines • Write the equation of the Line through (3, 2), which is perpendicular to 3x + 2y = -6.

38. Opener

39. Figures in the Plane Essential Question: How can slope, distance and/or midpoint be used to establish properties of a plane figure?

40. Parallelograms in Quadrilaterals? How should we draw it? • Origin & x-axis.

41. Labeling Vertices

42. Vertices of Parallelogram

43. Examples

44. Opener

45. Coordinate Proofs Essential Question: How can slope, distance and/or midpoint be used to establish properties of a plane figure?

46. Prove the Opener? • Can we PROVE that the midpoints of a Rhombus form a Rectangle? • Conveniently Plot a Rhombus, then find its midpoints…

47. Coordinate Proof:

48. PROVE Triangle Midsegment Theorem. • Midsegment is Parallel to Base and ½ the length of the 3rd side! • Place Triangle. Find Midpoints, calculate slope and length of Midsegment.

49. Plan out a Proof: