Calculate distance midpoint and slope between two points'

1
Chapter 13

• Calculate distance midpoint and slope between two
  points.
• Graph and find the equation for a line.
• Prove statements using coordinate geometry.

2
Lecture 1 (13-1)

• Objectives
• State and apply the distance formula
• State and apply the equation for a circle

3
Coordinate Plane
quadrants
I
II
origin
axes
0
IV
III
4
The Coordinates of a Point
The coordinates of a point are its address in the
plane.
A(2,2)
B(-1,1)
C(0,-2)
D(3,-2)
Sketch
5
The Distance Formula

• The distance between points can be calculated
  using the Pythagorean Theorem

(x1,y1)
d
y2-y1
x2-x1
(x2,y2)
Sketch
6
Examples

• Find the distance between the following pairs of
  points

A(3,0) B(0,-4)
C(0,2) D(-3,3)
E(3,-5) F(-2,-3)
7
Equation of a Circle

• Since a circle is the set of points equidistant
  from a given point, the distance formula can give
  the equation of a circle.

Sketch
8
Lecture 2 (13-2)

• Objectives
• Define and apply the slope of a line.

9
Slope

• The slope of a line indicates its direction.

(x1,y1)
y2-y1
x2-x1
(x2,y2)
Sketch
10
Examples

• Find the slope between the following pairs of
  points

A(3,0) B(0,0)
C(0,2) D(0,3)
E(3,-5) F(-2,-3)
11
Lecture 3 (13-3)

• Objectives
• Determine the slope relationships between lines
  that are parallel and perpendicular.

12
Parallel Lines

• Parallel lines have the same slope.

Sketch
13
Perpendicular Lines

• Perpendicular lines have slopes that multiply to
  -1.

Sketch
14
Lecture 4 (13-4)

• Objectives
• Define and understand the basic properties of
  vectors

15
Vector

• Any quantity that has both magnitude and
  direction. Things like force, velocity and
  acceleration are vector quantities.

16
Vector Notation

• Vectors are indicated using an ordered pair, like
  a point. The coordinates, however, are changes,
  not locations.

17
Magnitude of a Vector

• The magnitude of a vector is its length.

18
Slope of a Vector

• The slope of a vector is the ratio of its
  coordinates.

19
Scalar Multiple of a Vector

• Multiplying a vector by a real number k (called
  a scalar) changes the length of the vector. The
  direction is reversed if k lt 0.

20
Equal Vectors

• Vectors are equal if they have the same magnitude
  and direction (same length and slope)

21
Lecture 5 (13-5)

• Objectives
• Determine and apply the formula for the midpoint
  of a segment.

22
Midpoint

• If M is the midpoint of segment AB, then AM
  MB.

(1, 2)
M
23
Lecture 6 (13-6)

• Objectives
• Determine how to graph a linear equation.
• Understand and apply the forms of the equation of
  a line.

24
Equation for a Line

• Describes the set of all points that lie on a
  line.

Sketch
25
Graphing a Line

• To graph a line, determine 3 points that lie on
  the line and draw the graph.

x
y
0 4
2 0
1 2
26
Horizontal and Vertical Lines

• Horizontal lines have a formula of y a and
  vertical lines x b.

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Slope Intercept Form

• This form of the equation for a line is the
  easiest to write and graph.

(0, b)
Sketch
28
Graphing Slope Intercept Form
The y-intercept is (0, -2)
The slope says go over 3
and up one.
29
Lecture 7 (13-7)

• Objectives
• Write the equation for a line.

30
Writing the Equation for a Line

• The equation for a line can be written from
  almost any two pieces of information about the
  line.

Point and Slope
Two Points
Slope and Intercept
Two Intercepts
Homework
31
Point and Slope

• Write the equation for a line through (3, -2)
  with a slope of -1.

Start with slope intercept form.
Insert the point and slope.
Solve for b.
Rewrite the equation with m and b.
Return
32
Two Intercepts

• Write the equation for a line with x-intercept 3
  and y-intercept -2.

Write the points.
Find the slope.
Rewrite the equation with m and b.
Return
33
Two Points

• Write the equation for a line through (2, -1) and
  (-3, 4).

Find the slope.
Insert one point and slope.
Solve for b.
Rewrite the equation with m and b.
Return
34
Slope and Intercept

• Write the equation for a line with m 1/2 and
  with y-intercept of 7.

Start with slope intercept form.
Rewrite with m and b.
Return
35
Lecture 8 (13-8, 13-9)

• Objectives
• Determine the proper placement of points on the
  coordinate axes for a coordinate proof.
• Perform a coordinate proof.

36
Coordinate Proof

• Using distance, midpoint, slope and the equations
  for lines and circles to prove everything else we
  have learned in geometry.

Steps

• Locate the figure on the coordinate plane

• Calculate the distances, midpoints, etc

• Explain your conclusions

37
Example

• Prove that the diagonals of a rectangle are
  congruent.

38
Locate the Figure

• Determine the ideal placement of the figure in
  the coordinate plane so that each vertex is
  well-defined and as simple as possible.

(a, b)
(0, b)
(-c, d)
(c, d)
(a, 0)
(0, 0)
(-c, -d)
(c, -d)
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Calculate

• Determine which calculations need to be performed
  to complete the proof, then do them.

T
C
R
E
40
Explain

• Using words, explain why the calculations lead to
  the conclusion and form the basis for the proof.

Since the lengths of the two diagonals were
equal, they are congruent.