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

ExampleFind the distance

between points A(4, -2) and B(7, 2)

d = 5

13-2 TheoremAn equation of the circle

with center (a,b) and radius r is

r2 = (x – a)2 + (y-b)2

ExampleFind an equation of the circle with center (-2,5) and radius 3.

(x + 2)2 + (y – 5)2 = 9

ExampleFind the center and the radius of the circle with equation (x-1)2 + (y+2)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.

m = change in y-coordinate change in x-coordinate

Or m = rise

run

FORMULA FOR SLOPE

SLOPE OF A LINEm = 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

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

Slopes

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 = 147y = 2x - 5

Determine whether each pair of lines is parallel,

perpendicular, or neither

-5x + 3y = 23x – 5y = 15

Determine whether each pair of lines is parallel,

perpendicular, or neither

2x – 3y = 68x – 4y = 4

SECTION 13-4

Vectors

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

DEFINITIONS

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

Vector

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

DEFINITIONS

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

Given: Points P(-5,4) and Q(1,2)

Find PQFind PQ

EXAMPLE

In general, if the vector PQ = (a,b) then

kPQ = (ka, kb)

Scalar Multiple

Vectors having the same magnitude and the same

direction.

Equivalent Vectors

Two vectors are perpendicular if the

arrows representing them have perpendicular

directions.

Perpendicular Vectors

Two vectors are parallel if the arrows representing

them have the same direction or opposite

directions.

Parallel Vectors

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

EXAMPLE

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

EXAMPLE

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

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

ExampleFind the midpoint of

the segment joining the points (4, -6) and (-3, 2)

M(1/2, -2)

SECTION 13-6

Graphing Linear Equations

LINEAR EQUATIONis an equation whose

graph is a straight line.

The graph of any equation that can be written in the form

Ax + By = CWhere A and B are not both zero, is a line

13-6 Standard Form

ExampleGraph the line 2x – 3y = 12Find the x-intercept and

the y-intercept and connect to form a line

The slope of the line Ax + By = C (B ≠ 0) is

- A/BY-intercept = C/B

THEOREM

Theorem 13-7 Slope-Intercept form

y = mx + bwhere m is the slope and b is

the y -intercept

Write an equation of a line with the given y-intercept and slope

m=3 b = 6

SECTION 13-7

Writing Linear Equations

An equation of the line that passes through the point (x1, y1) and has slope m is

y – y1 = m (x – x1)

Theorem 13-8 Point-Slope Form

Write an equation of a line with the given slope and through a given point

m=-2P(-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