1-3 DISTANCE, MIDPOINT, AND SLOPE - salem.k12.va.us

Lesso

Form

Lesson 1

on 1-3

mulas

1-3: Formulas 1

The Coordinate Plan

Definition: In the coordinate plan
(called the x- axis)
(called the y- axis)
called the Origin.

Lesson 1

ne

ne, the horizontal number line
) and the vertical number line
) interest at their zero points

y - axis

Origin

x - axis

1-3: Formulas 2

The Distance Formul

The distance d between any two po
and (x2, y2 ) is given by the formula

Example: Find the distance be
x1 = -3, x2 = 4, y1 = 2

d = (−3 − 4)2 + (2 − 1
d = (−7)2 + (1)2 = 4
d = 50 or 5 2 or

Lesson 1

la

oints with coordinates (x1, y1)

a d = (x2 − x1)2 + ( y2 − y. 1)2

etween (-3, 2) and (4, 1)

2 , y2 = 1

1)2

49 + 1

r 7.07 3

1-3: Formulas

Midpoint Formula

In the coordinate plane, the coord

segment whose endpoints have co
+ +
are ⎛ x1 2 x2 , y1 2 y2 ⎞ .
⎝⎜ ⎠⎟

Example: Find the midpoint be

x1 = -2, x2 = 6, y1

M= ⎛ −2 + 6
⎝2

M= ⎛ 4 , 9⎞
⎝ 2 2⎠

Lesson 1

dinates of the midpoint of a

oordinates(x1, y1) and (x2 , y2 )

etween (-2, 5) and (6, 4)

y1 = 5, and y2 = 4

6 , 5 + 4⎞
2 ⎠

= ⎛ 2, 9⎞
⎝ 2⎠

1-3: Formulas 4

Slope Formula

Definition: In a coordinate plane,
its vertical rise over it

Formula: The slope m of a line c
coordinates(x1, y1) and
the formula m = y2 − y

x2 − x

Example: Find the slope betwee

x1 = −2, x2 = 4, y1

m = y2 − y1 = 5 −
x2 − x1 4 −

Lesson 1

, the slope of a line is the ratio of
ts horizontal run. rise

run

containing two points with

d (x2, y2 ) is given by

y1 where x1 ≠ x2.
x1

en (-2, -1) and (4, 5).

1 = −1, y2 = 5

− (−1) m = 6 =1
− (−2) 6

1-3: Formulas 5

Describing Lines

z Lines that have a positive slope
z Lines that have a negative slope
z Lines that have no slope (the slo

z Lines that have a slope equal to

Lesson 1

rise from left to right.
e fall from left to right.
ope is undefined) are vertical.

o zero are horizontal.

1-3: Formulas 6

Some More Examp

z Find the slope between (4,

m= −5 − −5 = 0 = 0
4−3 1

Since the slope is zero,

z Find the slope between (3,4

line. 4 − −2 = 6
3−3 0
m=

Since the slope is undefin

Lesson 1

ples

-5) and (3, -5) and describe it.

0

the line must be horizontal.
4) and (3,-2) and describe the

=∅

ned, the line must be vertical.

1-3: Formulas 7

Example 3 : Find the slope
given points and describe t

(7, 6) and (– 4, 6)

Solution: y2 − y1 up 0
m=
x2 − x1

= 6−6
(−4) − 7

=0
−11

=0
This line is horizontal.

Lesson 1

pe of the line through the
the line.

left 11
y (-11)

(– 4, 6) (7, 6)

x

1-3: Formulas 8

Example 4: Find the slope
given points and describe

(– 3, – 2) and (– 3, 8)

Solution: (– 3

m = y2 − y1

x2 − x1

= 8− (−2)
(−3) − (−3)
(– 3, –
= 10 undefined
0

This line is vertical.

Lesson 1

e of the line through the
e the line.

right 0

y

3, 8)

up 10 x
– 2)

1-3: Formulas 9

Practice

z Find the distance between (3, 2)
z Find the midpoint between (7, -2
z Find the slope between (-3, -1) a
z Find the slope between (4, 7) an
z Find the slope between (6, 5) an

Lesson 1

and (-1, 6).

2) and (-4, 8).

and (5, 8) and describe the line.

nd (-4, 5) and describe the line.

nd (-3, 5) and describe the line.

1-3: Formulas 10