CHAPTER 13
Geometry and
Algebra
SECTION 13-1
The Distance
Formula
Theorem 13-1
The distance between
two points (x1, y1) and
(x2, y2) is given by:
D = [(x2 – x1)2 + (y2-y1)2]½
Example
Find the distance
between points A(4, -2)
and B(7, 2)
d = 5
13-2 Theorem
An equation of the
circle with center
(a,b) and radius r is
2
2
2
r = (x – a) + (y-b)
Example
Find an equation of
the circle with center
(-2,5) and radius 3.
2
2
(x + 2) + (y – 5) = 9
Example
Find the center and
the radius of the circle
2
with equation (x-1) +
2
(y+2) = 9.
(1, -2), r = 3
SECTION 13-2
Slope of a Line
SLOPE
is the ratio of vertical
change to the horizontal
change. The variable m
is used to represent
slope.
FORMULA FOR SLOPE
m = change in y-coordinate
change in x-coordinate
m = rise
run
Or
SLOPE OF A LINE
m = y2 – y1
x2 – x1
HORIZONTAL LINE
a horizontal line containing
the point
(a, b) is described by the
equation y = b and has
slope of 0
VERTICAL LINE
a vertical line containing
the point (c, d) is described
by the equation x = c and
has no slope
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
SECTION 13-3
Parallel and
Perpendicular Lines
Theorem 13-3
Two nonvertical lines are
parallel if and only if their
slopes are equal
Theorem 13-4
Two nonvertical lines are
perpendicular if and only
if the product of their
slopes is - 1
Find the slope of a line
parallel to the line
containing points M and N.
M(-2, 5) and N(0, -1)
Find the slope of a line
perpendicular to the
line containing points
M and N.
M(4, -1) and N(-5, -2)
Determine whether each pair
of lines is parallel,
perpendicular, or neither
7x + 2y = 14
7y = 2x - 5
Determine whether each
pair of lines is parallel,
perpendicular, or neither
-5x + 3y = 2
3x – 5y = 15
Determine whether each
pair of lines is parallel,
perpendicular, or neither
2x – 3y = 6
8x – 4y = 4
SECTION 13-4
Vectors
DEFINITIONS
Vector– any quantity such
as force, velocity, or
acceleration, that has
both size (magnitude)
and direction
Vector
Vector AB is equal to the
ordered pair (change in
x, change in y)
DEFINITIONS
Magnitude of a vector- is
the length of the arrow
from point A to point B
and is denoted by the
symbol  AB 
Use the Pythagorean
Theorem or the Distance
Formula to find the
magnitude of a vector.
EXAMPLE
Given: Points P(-5,4) and
Q(1,2)
Find PQ
Find  PQ 
Scalar Multiple
In general, if the vector
PQ = (a,b)
then
kPQ = (ka, kb)
Equivalent Vectors
Vectors having the same
magnitude and the same
direction.
Perpendicular Vectors
Two vectors are
perpendicular if the
arrows representing
them have perpendicular
directions.
Parallel Vectors
Two vectors are parallel if
the arrows representing
them have the same
direction or opposite
directions.
EXAMPLE
Determine whether (6,-3)
and (-4,2) are parallel or
perpendicular.
EXAMPLE
Determine whether (6,-3)
and (2,4) are parallel or
perpendicular.
Adding Vectors
(a,b) + (c,d) = (a+c, b+d)
Find the Sum
Vector PQ = (4, 1) and
Vector QR = (2, 3). Find
the resulting Vector PR.
SECTION 13-5
The Midpoint Formula
Midpoint Formula
M( x1 + x2, y1 + y2)
2
2
Example
Find the midpoint of
the segment joining
the points (4, -6) and
(-3, 2)
M(1/2, -2)
SECTION 13-6
Graphing Linear
Equations
LINEAR EQUATION
is an equation whose
graph is a straight
line.
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
Example
Graph the line
2x – 3y = 12
Find the x-intercept and
the y-intercept and
connect to form a line
THEOREM
The slope of the line
Ax + By = C (B ≠ 0) is
- A/B
Y-intercept = C/B
Theorem 13-7 SlopeIntercept form
y = mx + b
where m is the slope and b
is the y -intercept
Write an equation of a
line with the given yintercept and slope
m=3 b = 6
SECTION 13-7
Writing Linear
Equations
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)
Write an equation of a
line with the given slope
and through a given
point
m=-2
P(-1, 3)
Write an equation of a
line with the through
the given points
(2, 5) (-1, 2)
Write an equation of a
line through (6, 4) and
parallel to the line
y = -2x +4
END