1. 13.1 The Distance Formulas

2. Review of Graphs

3. Coordinate Plane

4. Origin

5. Axes: x-axis y-axis

6. The arrowhead on each side represents the positive direction

7. Quadrants C II I
III IV

8. Remote Time
Which quadrant does the point lie in?

9. Remote Time
(3,1)

10. Remote Time
(-2,-1)

11. Remote Time
(5,-3)

12. Finding Distance A(3,4) B(-1,4) D(-3,-2) C(-3,1)
**Distance is always a positive Number**
A(3,4)
B(-1,4)
D(-3,-2)
C(-3,1)

13. Horizontal Distance AB = |3-(-1)| = 4 or AB = |-1-3| = 4 B(-1,4)
**A horizontal line will always have y-coordinates that are the same**
AB = |3-(-1)| = 4 or
AB = |-1-3| = 4
B(-1,4)
A(3,4)
What do you notice about the coordinates of each points?

14. **A vertical line will always have x-coordinates that are the same**
Vertical Distance
**A vertical line will always have x-coordinates that are the same**
C(-3,1)
CD = |1-(-2)|= 3 or
AB = |-2-1| = 3
D(-3,-2)

15. What do I do when two points do not lie on a horizontal or vertical line……..?
PARTNERS: Come up with a logical way in which you think you could find the distance of your slanted line.
A(1,2)
B(4,-2)

16. Make a right triangle
A(1,2)
C(1,-2)
B(4,-2)

17. Use the pythagorean theorem
C(1,-2)
B(4,-2)

18. Theorem
The Distance Formula: The distance d between two points is given by

19. Find the distance 6 13
4√5
(0,0) (0,-6)
(-3,-8) (2,4)
(9,-4) (1,8)

20. The equation of a circle
What would happened if I squared both sides of the distance formula?
d2 = (x2 – x1)2 + (y2 – y1)2
The equation of a circle with center (h, k) and a radius (r) is…
r2 = (x – h)2 + (y – k)2

21. 3 different ways to use equation of a circle
Write an equation given the center and radius
What are the coordinates of the center and the radius, given the equation
Write the equation given the center and a point

22. Whiteboards Brightstorm examples Classroom exercises pg. 526 #11 b

23. 13.2 Slope of a Line

24. Coach
Hud
Forrest
Chad Eggertson

25. Slope (m)
Steepness of a line

26. Slope: Pick 2 points on the line
(x1, y1) (x2, y2)
Does it matter which y you start with?

27. Partner Practice Find the slope of the line through (-3,4) and (-4,5)

28. 4 Types of Slope

29. “Read” the line Positive Negative
Partners: Come up with 2 points that would give me a positive slope and 2 points that would give me a negative slope.

30. No Slope Zero Slope Find the slope of CD: C (2,1) D (2, 7)
Find the slope of AB: A (1,2) B (5,2)

31. No Slope (undefined) Zero Slope

32. No Slope (undefined)
EC: Which axis has an undefined slope?

33. Zero Slope

34. White Board Practice
Find the slope of the line through (6,-3) and (-1,-2) . Then Graph it.
- 1/7
Find the slope of the line through (8,-4) and (-3,-4) . Then graph it.

35. White Board Practice
Sketch the line and find the coordinates of two other points on the line.
The line goes through P(-2,1) and has a
slope = 1 / 3

36. White Board Practice
Sketch the line and find the coordinates of two other points on the line.
The line goes through P(2,4) and has a
slope =

37. Remote Time
A: Always
B: Sometimes
C: Never
D: I don’t know

38. A: Always B: Sometimes C: Never D: I don’t know
The slope of a vertical lines is ___________ zero.

39. A: Always B: Sometimes C: Never D: I don’t know
The slope of a horizontal line is ___________ zero.

40. A: Always B: Sometimes C: Never D: I don’t know
The slope of a line that rises to the right is __________ positive

41. A: Always B: Sometimes C: Never D: I don’t know
The slope of a line that falls to the right is __________ negative

42. 13.3 Parallel and Perpendicular Lines

43. Graph the following Line AB A(1, -5) B (4,0)
A line with point at D (0, -3) and m = 5/3

44. Parallel Lines
Coplanar lines that do not intersect

45. Theorem
Two nonvertical lines are parallel if and only if their slopes are equal

46. Partners
Give me the coordinates of 2 points for one line and 2 points for another line
I want the lines to be parallel with negative slopes

48. Perpendicular lines

49. Graph the following Line AB A(-1, 1) B (4,-2)
A line with point at D (0, -3) and m = 5/3

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

51. In English… That means that the slopes of two perpendicular lines are
Opposite (one + and one -)
Reciprocals ( and )

52. White Board Practice
r || s and r  t
Slope of r
Slope of s
Slope of t

53. White Board Practice
r || s and r  t
Slope of r
Slope of s
Slope of t

54. White Board Practice
r || s and r  t
Slope of r
Slope of s
Slope of t

55. White Board Practice
r || s and r  t
Slope of r
Slope of s
Slope of t

56. White Board Practice
r || s and r  t
Slope of r
Slope of s
Slope of t

57. White Board Practice
r || s and r  t
Slope of r
Slope of s
Slope of t

58. Remote Time The slopes of two lines are given. Are the lines
A: Parallel
B: Perpendicular
C: Neither
D: I don’t know

59. A: Parallel B: Perpendicular C: Neither D: I don’t know

60. A: Parallel B: Perpendicular C: Neither D: I don’t know

61. A: Parallel B: Perpendicular C: Neither D: I don’t know

62. A: Parallel B: Perpendicular C: Neither D: I don’t know

63. A: Parallel B: Perpendicular C: Neither D: I don’t know

64. A: Parallel B: Perpendicular C: Neither D: I don’t know

65. A: Parallel B: Perpendicular C: Neither D: I don’t know

66. A: Parallel B: Perpendicular C: Neither D: I don’t know

67. A: Parallel B: Perpendicular C: Neither D: I don’t know

68. White Board Practice
Use slopes to show that a quadrilateral with vertices A(-2,7), B(3,7), C(6,11), and D(1,11) is a parallelogram.

69. Consider triangle ABC where A=(1,1) B=( 5,2 ) C=( 4,6 )
Do a quick sketch.
How could we prove that this triangle is an isosceles right triangle?
We’d need to:
Show two sides were congruent using the Distance Formula
Show that one of the angles is 90 degrees by Showing two segments are perpendicular (have slopes that are opposite reciprocals)

70. Consider triangle ABC where A=(1,1) B=( 5,2 ) C=( 4,6 )
Now Do It! Using the slope and distance formulas show that ABC is a right isosceles triangle.

71. 13-5 Midpoint Formula

72. Warm – up In your own words, write down what the midpoint formula is…
What is it allowing us to find?
How is the answer going to be given?

73. 4 facts from ch 13 Parallel lines have the same slope.
Perpendicular lines have opposite reciprocal slopes (2/3 vs. -3/2)

74. C A B Name the midpoint of segment AB.
Name the midpoint of segment AC.
Describe your strategy.
What does the midpoint of BC appear to be? Can you adapt your strategy above to find it algebraically?

75. Summary:
To find the midpoint of a segment, simply average the x coordinates and average the y coordinates.
OR:
The midpoint between
Is:_______________

76. 3 types of midpoint problems
Find the midpoint of segment
Find midpoint of AB if A=(-3, 7) and B=(9, -20)
Midpoint = (3, -6.5)
Find the other endpoint when given the midpoint
(3, -1) is the midpoint of segment RS. If R = (12, -5) find the coordinates of S.
S (-6, 3)

77. 3. Show a point is on the perpendicular bisector of a segment
Given points A(5,2) and B(3,-6) show that P(0,-1) is on the perpendicular bisector. Draw a sketch to help.
Find the midpoint of AB
Find slope of AB and slope of PM
If the product of slope AB and slope PM = -1, then P is on the perpendicular bisector of AB.

78. Whiteboards Finding other endpoint

79. C B A
Show that point C is on the perpendicular bisector of line AB.

80. Challenge: F=(-2,-4) G=(4,-1) H=(1,1) I=(-1,0) Sketch and label
A) Use the slope formula to show that FGHI is a trapezoid
B) Use the midpoint and slope formulas to show that the median of the trapezoid (the segment that connects the midpoints of the legs) is parallel to both bases.

81. 13.6 Graphing Linear Equations

82. What is a Linear Equation?y x
Linear
A linear equation is an equation whose graph is a LINE.
Not Linear

83. Does the point (4,2) lie on the line x + y = 6 ?

84. Forms of Linear Equations
The forms of linear equations are the formats in which the information is written in.
These two forms are the most commonly used ways to write linear equations.
1. Standard Form: Ax + By =C
2. Slope Intercept Form: y=mx+b

85. Standard form for linear equations is Ax + By=C.
It can be used to find two points on the line of the equation.
**More specifically the x and y int.**
Example: 2x+2y=4
1.) Substitute in a zero for x. Simplify.
2(0)+2y=4
y=2
2.) One point of the line is (0,2). Plot the point.
3.) Substitute y with zero. Simplify.
2x+2(0)=4
x=2
4.) The second point is (2,0). Plot it and draw a line through the two points. y x

86. Slope Intercept Form *Get y by itself*
Slope intercept form is y=mx+b.
This form makes it easy to find the slope (m) and the y-intercept (b).
Working with this form is simple, so it is used more often than other forms.
Example: y= ¾x + 3
* ¾ is the slope.
* 3 is the point where the line crosses the Y-axis.

87. Graph Using Slope Intercept Form
I will use the equation
y= ½ x-1 to demonstrate how
to graph a linear equation
using slope intercept form.
1.) First, graph the y-intercept (b). In this equation, b= -1, so place a dot on the point (0,-1).
2.) Next, use the slope (½ ) to rise up once and run twice horizontally. Plot the point.
3.) Plot as many points as desired using the slope, then draw a line through the points.
Run 2
Rise 1

88. Graphing a Linear Equationy x
Let’s try x – 2y = 6.
First Step:
Write y as a function of x
x – 2y = 6
–2y = 6 – x 3
y = 1/2x - 3

89. Graphing Horizontal & Vertical Linesy x
When you are asked to graph a line, and there is only ONE variable in the equation, the line will either be vertical or horizontal. For example …
THINK OPPOSITE OF THE AXIS!!
Graph x = 3
Since there are no y – values in this equation, x is always 3 and y can be any other real number.
y = –2
Graph y = –2
Since there are no x – values in this equation, y is always – 2 and x can be any other real number.
x = 3

90. Solving System of Equations
2x + 3y = 18
2x - y = 10
2 different ways to solve…
Substitution Method
Add/subtract method
PARTNERS: find the value of x and y using one of the methods.
**What is the x and y values that will work for both equations?
(In other words….where on the coordinate plane do the two lines intersect?)

91. Whiteboards
Page 69:
#4,
11, 18
Pg. 550 (classroom exercises) #5
#11

92. 13.7 Writing Linear Equations

93. WARM -UP Use pg. 553 to help answer to following…
Put the equation of the line in slope intercept form
m = -5/3 ; y-int = 4
x-int = -6 ; y-int = 3

94. Forms of Linear Equations
Standard Form: Ax + By =C
Slope Intercept Form: y=mx+b
Point-Slope Form: y – y1 = m(x-x1)
**Always simplify the final answer by putting it in slope-intercept form.**

95. y – y1 = m(x - x1) Point Slope Form
Best used for writing the equation of a line when given information about the line.
It will helps us get to the point where we can write the equation in slope int. form
An equation of the line that passes through the point (x1, y1) and has slope m.
y – y1 = m(x - x1)

96. Find an equation of the line described
Ex1. The line with slope 2 containing the point (5,-2)
y – y1 = m(x - x1)
y + 2 = 2(x – 5)  insert m and the point
y + 2= 2x –  distribute the 2 to the x and -5
y= 2x –  get y by itself

97. y + 2 = 3(x-3)  plug in m and 1 point y = 3x – 11  slope int form
Find an equation of the line described
Ex 2. The line through (3,-2) and (4,1)
-2 – m =  Find the slope
3-4
y + 2 = 3(x-3)  plug in m and 1 point
y = 3x –  slope int form

98. y = 3/5x - 3 Find an equation of the line described
Ex3. The line with x-intercept 5 and y-intercept –3.
y = 3/5x - 3

99. Find an equation of the line described
Ex4. The line through (6,4) and perpendicular to the line y = -2x+4
y – 4 = ½(x - 6)
y = 1/2x + 1

100. x = 5 Find an equation of the line described
Ex 5. A vertical line going through (5,-3)
x = 5

101. y = 7 Find an equation of the line described
Ex 5. A horizontal line going through (-4,7)
y = 7

102. Organizing Coordinate Proofs
Chapter 13.8
Organizing Coordinate Proofs

103. Review Squares
Four congruent sides
Four Right angles

104. Review Rectangles
Opposite sides are congruent
Four Right angles

105. Review Parallelograms
Opposite sides are congruent
Opposite angles are congruent
Opposite sides are parallel

106. Review Isosceles Triangles
Legs are congruent
Base angles are congruent

107. Review Isosceles Trapezoids
Legs are congruent
Base angles are congruent

108. Review Rhombus
All sides are congruent

109. Quick Review
In an ordered pair which one is the x-coor and which is the y-coor?
The x-coordinate tells us to do what ?
The y-coordinate tells us to do what?
(3,-7)

110. Rectangle
( 4 ,3 )
( , )
( , )
( , )

111. Rectangle
( a ,b )
( , )
( , )
( , )

112. Equilateral Triangle
( 3 , 5 )
( , )
( , )

113. Equilateral Triangle
( a , b )
( , )
( , )

114. Rhombus
( , )
( a , b )
( , )
( c , 0 )

115. Square
( , )
( , )
( , )
( 3k , 0 )

116. Isosceles Triangle
( , b )
( , )
( 2a , 0 )

117. Isosceles Trapezoid
( , )
( 2e , f )
( -3e ,0 )
( 3e , 0 )

118. Isosceles Right Triangle
( , )
( , )
( s , 0 )

119. Parallelogram
( h , )
( , g )
( i, 0 )