Lesson 1-3: Formulas 1

Lesson 1-3

Formulas

Lesson 1-3: Formulas 2

The Coordinate PlaneIn the coordinate plane, the horizontal number line

(called the x- axis) and the vertical number line (called the y- axis) interest at their zero points called the Origin.

Definition:

x - axis

y - axis

Origin

Lesson 1-3: Formulas 3

The Distance Formula

Find the distance between (-3, 2) and (4, 1)

1 1( , )x y

x1 = -3, x2 = 4, y1 = 2 , y2 = 1

( 3 4)2 (2 1)2d =

( 7)2 (1)2 49 1d =

50 or 5 2 or 7.07d =

The distance d between any two points with coordinates and is given by the formula d = .2 2( , )x y 2 2

2 1 2 1( ) ( )x x y y

Example:

Lesson 1-3: Formulas 4

Midpoint Formula

1 2 1 2,2 2

x x y y1 1( , )x y

42

,92

= 2,

92

M =

2 62

,5 4

2

M =

Find the midpoint between (-2, 5) and (6, 4)

x1 = -2, x2 = 6, y1 = 5, and y2 = 4

Example:

In the coordinate plane, the coordinates of the midpoint of a segment whose endpoints have coordinates and are .

2 2( , )x y

Lesson 1-3: Formulas 5

Slope Formula

rise

run

Find the slope between (-2, -1) and (4, 5).Example:

Definition: In a coordinate plane, the slope of a line is the ratio of its vertical rise over its horizontal run.

Formula:

1 1( , )x y 2 2( , )x y

The slope m of a line containing two points with

coordinates and is given by

the formula where . 2 1

2 1

y ym

x x

1 2x x

1 2 1 22, 4, 1, 5x x y y 2 1

2 1

5 ( 1)

4 ( 2)

y ym

x x

6

16

m

Lesson 1-3: Formulas 6

Describing Lines

Lines that have a positive slope rise from left to right.

Lines that have a negative slope fall from left to right.

Lines that have no slope (the slope is undefined) are vertical.

Lines that have a slope equal to zero are horizontal.

Lesson 1-3: Formulas 7

Some More Examples

Find the slope between (4, -5) and (3, -5) and describe it.

Since the slope is zero, the line must be horizontal.

5 54 3

01

0m =

Find the slope between (3,4) and (3,-2) and describe the line. 4 2

3 3

60

m =

Since the slope is undefined, the line must be vertical.

Lesson 1-3: Formulas 8

Example 3 : Find the slope of the line through the given points and describe the line.

(7, 6) and (– 4, 6)

2 1

2 1

6 6

( 4) 7

0

110

y y

x x

Solution:

This line is horizontal.

x

y

(7, 6)

up 0

left 11 (-11)

(– 4, 6)

m

Lesson 1-3: Formulas 9

Example 4: Find the slope of the line through the given points and describe the line.

(– 3, – 2) and (– 3, 8)

2 1

2 1

8 2

( 3) 3

10

0

y y

x x

Solution:

This line is vertical.

x

y

(– 3, – 2)

up 10

right 0

(– 3, 8)

undefined

m

Lesson 1-3: Formulas 10

Practice

Find the distance between (3, 2) and (-1, 6).

Find the midpoint between (7, -2) and (-4, 8).

Find the slope between (-3, -1) and (5, 8) and describe the line.

Find the slope between (4, 7) and (-4, 5) and describe the line.

Find the slope between (6, 5) and (-3, 5) and describe the line.