1. Write an equation given the following info:
Warm up
Write an equation given the following info:
1. m = (-9, -1) (-2, -1) (-2, 3)
3. (-2, 6) (2, 8) m = 0 (3, 4)

2. 5. & 6. Write the equation of each graph   
y = 2x – 3
y = -3/2x + 3

4. Unit 2: Geometric & Algebraic Connections
LG 2-1 Coordinate Proofs LG 2-2 Equations of Circles LG 2-3 Geometric Modeling TEST February 7th!

5. LG 2-1 Coordinate Proofs:
write the equations of parallel and perpendicular lines
find the point that partitions a line segment in a given ratio
use coordinate algebra to determine perimeter and area of defined figures
ESSENTIAL QUESTIONS
How can a line be partitioned?
How are the slopes of lines used to determine if the lines are parallel, perpendicular, or neither?
How can slope and the distance formula be used to determine properties of polygons and classify them?

6. SELECTED TERMS AND SYMBOLS
Distance Formula
Formula for finding the point that partitions a directed segment AB at the ratio of a : b from A(x1, y1) to B(x2, y2):

7. Writing Equations of Lines

8. I. How to Write an Equation of a Line Given m and b
1. Write down y = mx + b
2. Substitute slope for m and y-intercept for b.
3. Simplify the equation

9. Write the equation of the line given m and b.
Ex. 1 Slope is -5 and y-intercept is 2
y = -5x + 2
Ex. 2 Slope is -1/2 and y-intercept is -2
y = -½x – 2

10. Write the equation of the line given m and b.
Ex. 3 Slope is 0 and y-intercept is 3
y = 3
Ex. 4 Slope is 1/3 and y-intercept is 0

11. II. How to Write an Equation of a Line Given a Graph
Write down y = mx + b
Use any 2 “good” points on the graph to find the slope, m.
Find the y-intercept on the graph, b.
Substitute slope for m and y-int for b into the equation y = mx + b.

12. 5. Write the equation of this graph 

13. 6. Write the equation of this graph 

14. 7. Write the equation of this graph 
y = -3x + 2

15. 8. Write the equation of this graph 
y = 2x

16. 9. Write the equation of this graph 
y = 3

17. 10. Write the equation of this graph
x = - 6

18. 11. & 12. Write the equation of each graph   
y = 2x – 3
y = -3/2x + 3

19. III. How to Write an Equation of a Line Given m and a point
Write down y = mx + b.
Substitute slope for m and the point (x, y).
Solve for b.
Substitute m and b back into the equation.

20. Write the equation of the line given m and a point
Ex 13: m = 2 Point: (2, 3)
y = mx + b
3 = 2 (2) + b
b = -1
y = 2x – 1

21. Write the equation of the line given m and a point
Ex 14: m = 1/2 Point: (4,-3)
y = mx + b
-3 = 1/2 (4) + b
b = -5
y = ½ x – 5

22. Ex 15: m = -2 Point: (-5, 3) y = mx + b 3 = -2 (-5) + b b = -7
Write the equation of the line given m and a point
Ex 15: m = -2 Point: (-5, 3)
y = mx + b
3 = -2 (-5) + b
b = -7
y = -2x – 7

23. Write the equation of the line given m and a point
Ex: 16
m = 4 (1,4)
y = 4x

24. Write the equation of the line given m and a point
Ex: 17
m = ½ (-1,-2)
y = ½ x – 1 ½

25. Write the equation of the line given m and a point
Ex: 18
m = 2 (0,3)
y = 2x + 3

26. Write the equation of the line given m and a point
Ex: 19
m = 3 (3,0)
y = 3x – 9

27. Write the equation of the line given m and a point
Ex: 20
m = undefined (3,6)
x = 3

28. IV. How to Write an Equation of a Line Given TWO points
Write down y = mx + b.
Use the slope formula to find m.
Pick one of the ordered pairs & substitute slope for m and the point (x, y).
Solve for b.
Substitute m and b into the equation.

29. Equation of a Line - Given 2 points
Ex: (2, 3) (4, 5)
y = mx + b
3 = 1(2) + b
b = 1
y = x + 1

30. Equation of a Line - Given 2 points
Ex: (2, 3) (-4, 15)
y = -2x + 7

31. Equation of a Line - Given 2 points
Ex: (2, 2) (0, 4)
y = -x + 4

32. Equation of a Line - Given 2 points
Ex: 24
(2,3) (1,4)
y = -x + 5

33. Equation of a Line - Given 2 points
Ex: 25
(4,5) (5,2)
y = -3x + 17

34. Graphs: Lines Never Intersect and are in the same plane
PARALLEL LINES
Graphs: Lines Never Intersect and are in the same plane
Equations:
Same Slopes
Different y-intercepts

35. How to Write an Equation of a Line PARALLEL to another and given a point
1. Given equation should be solved for y (y = mx + b)
Write down the slope of that line
Substitute m and (x, y) in y = mx + b.
Solve for b.
Write the equation using m and b.

36. Write a line parallel to the line 2x + y = 3 and passes through the point (-2, 5).
Work on left side of t-chart, step by step notes on right side
y = -2x + 1

37. Write a line parallel to the line y = 3x – 5 and passes through the point (-5, -2).
Work on left side of t-chart, step by step notes on right side
y = 3x + 13

38. Write a line parallel to the line y = -4x + 1 and passes through the point (2, -1).
Work on left side of t-chart, step by step notes on right side
y = -4x + 7

39. Write a line parallel to the line y = -x – 7 and passes through the point (-4, -4).
Work on left side of t-chart, step by step notes on right side
y = -x – 8

40. Graphs: Lines Intersect at 90º angles
PERPENDICULAR LINES
Graphs: Lines Intersect at 90º angles
Equations:
Opposite Reciprocal Slopes
With the same or different y-int

41. Find the Opposite Reciprocal Slopes

42. Are these lines parallel, perpendicular, or neither?
y = -2x + 1
y = -2x – 4
y = 3x – 4
y = -3x + 1
y = 1/5 x + 2
y = -5x + 6
parallel
neither
perpendicular

43. Are these lines parallel, perpendicular, or neither?
4. y = -2x + 1
y = -1/2x – 4
5. y = 3x – 4
y = 1 + 3x
6. y = 5/6 x + 2
y = -6/5 x + 6
neither
parallel
perpendicular

44. 1. Given equation should be solved for y (y = mx + b)
How to Write an Equation of a Line PERPENDICULAR to another and given a point
1. Given equation should be solved for y (y = mx + b)
Write down the OPPOSITE RECIPROCAL slope of that line
Substitute m and (x, y) in y = mx +b.
Solve for b.
Write the equation using m and b.

45. Write a line perpendicular to the line y = ½ x – 2 and passes through the point (1, 0).
Work on left side of t-chart, step by step notes on right side
y = -2x + 2

46. Write a line perpendicular to the line y = -3x + 2 and passes through the point (6, 5).
Work on left side of t-chart, step by step notes on right side
y = 1/3x + 3

47. Write a line perpendicular to the line 2x + 3y = 9 and passes through the point (6, -1).
Work on left side of t-chart, step by step notes on right side
y = 3/2 x – 10

48. Write a line perpendicular to the line y = 2x – 1 and passes through the point (2, 4).
Work on left side of t-chart, step by step notes on right side
y = -1/2x + 5

49. Write a line perpendicular to the line and passes through the point (5, 1).
Work on left side of t-chart, step by step notes on right side
y = 3x – 14

50. Determine whether the given lengths are sides of a right triangle.
Warm-up
Use the Pythagorean theorem to find the missing length of the right triangle. Round to the nearest tenth.
Determine whether the given lengths are sides of a right triangle.
3. 8, 15, , 6, , 40, 41 c 2 17 8 5 b

51. Write the following equations:
A line parallel and a line perpendicular to the line 3x – 2y = 8 through the point (-2,1).
A line parallel and a line a line perpendicular to the line x = 7 through the point (3, 8).
A line parallel and a line perpendicular to the line y = 7 through the point (-4, 0).

52. The Distance Formula
(x2,,y2) d
(x1,,y1)
(x2,,y1)

53. Warm UP

54. The Distance Formula

55. Steps to solving the Distance Formula
Write the distance formula
Substitute
Simplify
Evaluate Powers
Add
Use a calculator

56. The Distance Formula
(3,4)
(1,1)

57. Example #1
Use the distance formula to find the distance between the points, (10, 5) and (40, 45).
D = 50

58. Use the distance formula to find the distance
Example #2
Use the distance formula to find the distance
between (1, 4) and (-2, 3)
D = 3.16

59. The midpoint between (x1, y1) and (x2, y2) is
The Midpoint Formula
The midpoint of a line segment is the point on the segment that is equidistant from its endpoints
The midpoint between (x1, y1) and (x2, y2) is

60. Example 1
Find the midpoint of the line segment connecting the given points. (-2, 3) and (4, 2)

61. Example 2
Find the midpoint of the line segment connecting the given points. (200, 75) and (25, 175).

62. Find the midpoint of the line segments.
Example
Find the midpoint of the line segments.

63. Partition Line Segments (1 Dimension)
A is at x1, and B is at x2. Find the point, T, so that T partitions A to B in an a:b ratio.

64. 1. A is at 1, and B is at 7. Find the point, T, so that T partitions A to B in a 2:1 ratio.

65. 2. A is at -6 and B is at 4. Find the point, T, so that T partitions A to B in a 2:3 ratio.

66. Partition Line Segments (2 Dimensions)
A is at (x1, y1) and B is at (x2, y2) Find the point, T, so that T partitions A to B in an a:b ratio.

67. Ex 1: Find the coordinates of P along the directed line segment AB so that the ratio of AP to PB is 3 to 2.
(4.8, 7.6)

68. Ex 2: Find the coordinates of point P along the directed line segment AB so that AP to PB is the given ratio. A(1, 3), B(8, 4); 4 to 1.
(6.6, 3.8)

69. EX 3: Find the coordinates of point P along the directed line segment AB so that AP to PB is the given ratio. A(-2, 1), B(4, 5); 3 to 7.
(-0.2, 2.2)

70. Ex 4: Find the coordinates of point P along the directed line segment AB so that AP to PB is the given ratio. A(8, 0), B(3, -2); 1 to 4.
(7, -0.4)

71. Review
Find the coordinates of point P along the directed line segment AB so that AP to PB is the given ratio. A(-2, -4), B(6, 1); 3 to 2.
Find the coordinates of point P along the directed line segment BA in a 1:4 ratio. A(8, 0), B(3, -2)

72. Connecting Geometric & Algebraic Concepts:
The goal of this assignment is to use the distance and slope formulas to prove statements about geometric figures on the coordinate plane. Since the purpose is to prove a statement, you must show work.

73. Quadrilateral 1: Plot and label each point
Quadrilateral 1: Plot and label each point. A(-5, 6), B(3, 7), C(4, -1), and D(-4, -2).
Definition: A parallelogram is a quadrilateral with two pairs of opposite sides that are parallel. Using the definition of parallelogram, prove that Quadrilateral 1 is a parallelogram

74. Quadrilateral 1: Plot and label each point
Quadrilateral 1: Plot and label each point. A(-5, 6), B(3, 7), C(4, -1), and D(-4, -2).
Theorem: A parallelogram with four right angles is a rectangle. Using the theorem, prove that Quadrilateral 1 is a rectangle.

75. Quadrilateral 1: Plot and label each point
Quadrilateral 1: Plot and label each point. A(-5, 6), B(3, 7), C(4, -1), and D(-4, -2).
Definition: A rhombus is a parallelogram with all sides congruent. Using the definition, prove that Quadrilateral 1 is a rhombus.

76. Quadrilateral 1: Plot and label each point
Quadrilateral 1: Plot and label each point. A(-5, 6), B(3, 7), C(4, -1), and D(-4, -2).
Definition: A square is a rectangle and a rhombus. Using the definition, is Quadrilateral 1 a square? Why?

77. Quadrilateral 1: Plot and label each point
Quadrilateral 1: Plot and label each point. A(-5, 6), B(3, 7), C(4, -1), and D(-4, -2).
Theorem: The diagonals in a rhombus are perpendicular. Prove that the theorem is true for Quadrilateral 1.

78. Continue working on the packet 

80. Warm UP

81. LG 2-2 Equations of Circles:
Develop equations from geometric definition of circles
address equations of circles in standard and general forms
graph circles by hand and by using graphing technology
ESSENTIAL QUESTIONS
How can I use the Pythagorean Theorem to derive the equation of a circle?
How are the graph of a circle and its equation related?
How are the equation of a circle and its graph related?
How can I prove properties of geometric figures algebraically?

82. SELECTED TERMS AND SYMBOLS
Center of a Circle: The point inside the circle that is the same distance from all of the points on the circle.
Standard Form of a Circle:
where (h,k) is the center and r is the radius.

84. For this task, you need a straightedge (a ruler), a PENCIL and a calculator.

85. Graphing and Writing Equations of Circles

86. Standard Form of a Circle r is the radius of the circle
Center is at (h, k)
r is the radius of the circle

87. EX 1 Write an equation of a circle with center (3, -2) and a radius of 4.k r

88. EX 2 Write an equation of a circle with center (-4, 0) and a diameter of 10.k 2r

89. EX 3 Write an equation of a circle with center (2, -9) and a radius of .k r

90. EX 4 Find the coordinates of the center and the measure of the radius.
Opposite signs!
( , ) 6 -3
Take the square root!
Radius 5

91. 5. Find the center, radius, & equation of the circle.
The center is
The radius is
The equation is
(0, 0) 12
x2 + y2 = 144

92. 6. Find the center, radius, & equation of the circle.
(1, -3)
The center is
The radius is
The equation is 7
(x – 1)2 + (y + 3)2 = 49

93. 7. Graph the circle, identify the center & radius.
(x – 3)2 + (y – 2)2 = 9
Center (3, 2)
Radius of 3     

94. Warm UP
1) Write the equation of a circle that passes through (2, 3) and has center Q(2, –1) 2) Write the equation of a circle that passes through (–2, –1) and has center D(2, –4).

95. General Form of a Circle

96. General Form of a Circle
Every binomial squared has been multiplied out.
Every term is on the left side, equal to 0.
Squared terms go first in alpha order.

97. Converting from General to Standard
A needs to be 1. Divide if needed.
Move the x terms together and the y terms together.
Move E to the other side of the equals sign.
Complete the square (as needed) for x.
Complete the square(as needed) for y.
Factor the left & simplify the right.

98. 8. Write the standard equation of the circle. State the center & radius.

99. 9. Write the standard equation of the circle. State the center & radius.

100. 10. Write the standard equation of the circle
10. Write the standard equation of the circle. State the center & radius.

101. 11. Write the general form of the equation of the circle.

102. Review Test

103. NOPE! Warm UP! 1) Write the equation of the circle:
(x – 2)2 + (y + 4)2 = 16
NOPE!
3) A point on the circle has
coordinates (3,y). Find a value
for y.
y = 4 or -4

104. In your own words, compare and contrast the following circles
In your own words, compare and contrast the following circles. Write in complete sentences. Turn this in when you are finished.

105. Write equations for the following circles (if you didn’t already)
Write equations for the following circles (if you didn’t already). What do you call the FORM of the equations you wrote?

106. ODD ONE OUT!
Each of these three circles might be considered to be the odd one out.
With a partner, choose one and write why it might be the odd one out.
What properties have the other two circles got in common that the third does not have?

107. Write the NUMBER of the card on the placemat 
group
Write the NUMBER of the card on the placemat 
DO NOT WRITE ON THE CARDS PLEASE!

110. On the back of your quiz:
Extra Credit
On the back of your quiz:

113. Warm UP!

114. QUIZ TIME

116. Prove or disprove that the point A(10, 3) lies on a circle centered at C(5, -2) and passing through the point B(6, 5).

117. To add points to your quiz yesterday, complete the following
To add points to your quiz yesterday, complete the following. Each question replaces one you missed.
Write the equation of the line through the point (3,3) for the line perpendicular to y = -1/3x + 5.
2. Write the equation of the line through the point (0, 7) for the line parallel to 3x + 4y = 12.
3. Given the directed segment from A(5,4) to B(0, 14) partition the segment with point P in a ratio of 2:3.

119. Warm UP:

120. LG 2-3 Geometric Modeling:
use Algebra to model Geometric ideas
Solve real world problems that can be modeled using density, area, and volume concepts.
Model everyday objects using three dimensional shapes and describe the object using characteristics of the shape.
ESSENTIAL QUESTIONS
How can I minimize cost and maximize the volume of a topless box?

121. LG 2-3 Geometric Modeling
Geometric Modeling is the field that discusses the mathematical methods behind the modeling of realistic objects. One of the main applications of this field is for computer graphics and computer aided design. The field was initially developed in the mid-1970s and has evolved as the speed and memory of computer systems has advanced.

122. Find the intersection of the two equations:

123. Josh and Drake decide to play catch after school
Josh and Drake decide to play catch after school. They start at the same point. Josh walks 50 feet north and 20 feet west. Drake walks 40 feet south and 10 feet east. How far apart are they?
Joe’s Pizza Parlor advertises free delivery within a 15 mile radius. If a customer lives 12 miles east and 10 miles south of Joe’s, do they qualify for free delivery?
A given circle has a radius of 6 and is centered at the origin. The line x = 3 intercepts the circle and forms a chord. What is the length of the chord? Round your answer to the nearest tenth.