1-3 Distance, Midpoint, And Slope[1]

8/6/2019 1-3 Distance, Midpoint, And Slope[1]

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Lesson 1-3: Formulas 1

Lesson 1-3

Formulas


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The Coordinate Plane

In the coordinate plane, the horizontal number line

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

Definition:

x - axis

y - axis

Origin


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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 dbetween any two points with coordinatesand is given by the formula d= .2 2( , )x y 2 2

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

Example:


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Midpoint Formula

1 2 1 2,2 2

+ +

x x y y1 1( , )x y

4

2,9

2

= 2,

9

2

M =

2+ 6

2 ,

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 andare .

2 2( , )x y


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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 ofits 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 = =


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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.


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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 5

4 3=

0

1= 0m =

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

line. 4 2

3 3=

6

0=

m =

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


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

(7, 6) and ( 4, 6)

2 1

2 1

6 6

( 4) 7

0

11

0

=

=

=

=

y y

x x

Solution:

This line is horizontal.

x

y

(7, 6)

up 0

left 11(-11)

( 4, 6)

m


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Example 4: Find the slope of the line through thegiven 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


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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.

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1-3 Distance, Midpoint, And Slope[1]