1. CHAPTER 13
Geometry and Algebra

2. SECTION 13-1
The Distance Formula

3. The distance between two points (x1, y1) and (x2, y2) is given by:
Theorem 13-1
The distance between two points (x1, y1) and (x2, y2) is given by:
D = [(x2 – x1)2 + (y2-y1)2]½

4. Example
Find the distance between points A(4, -2) and B(7, 2)
d = 5

5. An equation of the circle with center (a,b) and radius r is
13-2 Theorem
An equation of the circle with center (a,b) and radius r is
r2 = (x – a)2 + (y-b)2

6. Example
Find an equation of the circle with center (-2,5) and radius 3.
(x + 2)2 + (y – 5)2 = 9

7. Example
Find the center and the radius of the circle with equation (x-1)2 + (y+2)2 = 9.
(1, -2), r = 3

8. SECTION 13-2
Slope of a Line

9. SLOPE
is the ratio of vertical change to the horizontal change. The variable m is used to represent slope.

10. FORMULA FOR SLOPE Or m = rise run m = change in y-coordinate
change in x-coordinate Or
m = rise
run

11. SLOPE OF A LINE
m = y2 – y1
x2 – x1

12. a horizontal line containing the point
(a, b) is described by the equation y = b and has slope of 0

13. VERTICAL LINE
a vertical line containing the point (c, d) is described by the equation x = c and has no slope

14. Lines with positive slope rise to the right.
Slopes
Lines with positive slope rise to the right.
Lines with negative slope fall to the right.
The greater the absolute value of a line’s slope, the steeper the line

15. Parallel and Perpendicular Lines
SECTION 13-3
Parallel and Perpendicular Lines

16. Theorem 13-3
Two nonvertical lines are parallel if and only if their slopes are equal

17. Theorem 13-4
Two nonvertical lines are perpendicular if and only if the product of their slopes is - 1

18. Find the slope of a line parallel to the line containing points M and N.

19. Find the slope of a line perpendicular to the line containing points M and N.

20. Determine whether each pair of lines is parallel, perpendicular, or neither
7x + 2y = 14
7y = 2x - 5

21. Determine whether each pair of lines is parallel, perpendicular, or neither
-5x + 3y = 2
3x – 5y = 15

22. Determine whether each pair of lines is parallel, perpendicular, or neither
2x – 3y = 6
8x – 4y = 4

23. SECTION 13-4
Vectors

24. DEFINITIONS
Vector– any quantity such as force, velocity, or acceleration, that has both size (magnitude) and direction

25. Vector AB is equal to the ordered pair (change in x, change in y)

26. DEFINITIONS
Magnitude of a vector- is the length of the arrow from point A to point B and is denoted by the symbol  AB 

27. Use the Pythagorean Theorem or the Distance Formula to find the magnitude of a vector.

28. Given: Points P(-5,4) and Q(1,2)
EXAMPLE
Given: Points P(-5,4) and Q(1,2)
Find PQ
Find  PQ 

29. In general, if the vector PQ = (a,b)
Scalar Multiple
In general, if the vector PQ = (a,b)
then
kPQ = (ka, kb)

30. Vectors having the same magnitude and the same direction.
Equivalent Vectors
Vectors having the same magnitude and the same direction.

31. Perpendicular Vectors
Two vectors are perpendicular if the arrows representing them have perpendicular directions.

32. Parallel Vectors
Two vectors are parallel if the arrows representing them have the same direction or opposite directions.

33. Determine whether (6,-3) and (-4,2) are parallel or perpendicular.
EXAMPLE
Determine whether (6,-3) and (-4,2) are parallel or perpendicular.

34. Determine whether (6,-3) and (2,4) are parallel or perpendicular.
EXAMPLE
Determine whether (6,-3) and (2,4) are parallel or perpendicular.

35. Adding Vectors
(a,b) + (c,d) = (a+c, b+d)

36. Find the Sum
Vector PQ = (4, 1) and Vector QR = (2, 3). Find the resulting Vector PR.

37. SECTION 13-5
The Midpoint Formula

38. Midpoint Formula
M( x1 + x2, y1 + y2)

39. Example
Find the midpoint of the segment joining the points (4, -6) and (-3, 2)
M(1/2, -2)

40. Graphing Linear Equations
SECTION 13-6
Graphing Linear Equations

41. is an equation whose graph is a straight line.
LINEAR EQUATION
is an equation whose graph is a straight line.

42. The graph of any equation that can be written in the form
13-6 Standard Form
The graph of any equation that can be written in the form
Ax + By = C
Where A and B are not both zero, is a line

43. Example Graph the line 2x – 3y = 12
Find the x-intercept and the y-intercept and connect to form a line

44. The slope of the line Ax + By = C (B ≠ 0) is - A/B Y-intercept = C/B
THEOREM
The slope of the line
Ax + By = C (B ≠ 0) is
- A/B
Y-intercept = C/B

45. Theorem 13-7 Slope-Intercept form
y = mx + b
where m is the slope and b is the y -intercept

46. Write an equation of a line with the given y-intercept and slope
m=3 b = 6

47. Writing Linear Equations
SECTION 13-7
Writing Linear Equations

48. Theorem 13-8 Point-Slope Form
An equation of the line that passes through the point (x1, y1) and has slope m is
y – y1 = m (x – x1)

49. Write an equation of a line with the given slope and through a given point
m=-2
P(-1, 3)

50. Write an equation of a line with the through the given points
(2, 5) (-1, 2)

51. Write an equation of a line through (6, 4) and parallel to the line y = -2x +4

52. END