Download Distance Formula

Goal 1
Find the Midpoint of a Segment
Goal 2
Find the distance between two
points on a coordinate plane
Goal 3
Find the slope of a line between two
points on a coordinate plane
Distance Formula
Used to find the distance between two points
( x1 , y1 )and ( x2 , y2 )
distance  ( x2  x1 )  ( y2  y1 )
2
2
Example
 Find the distance between (2,1)
and (5,2).
x1 y1
 D= (2 - 5)² + (1 - 2)²
 D= (-3)² + (-1)²
 D=
9+1
 D= 10
 D= 3.162
x2 y2
-First substitute numbers for
variables and solve the
parentheses.
-Then solve the squared number.
-Add the two numbers.
-Find the square root of the
remaining number.
Example
Find the distance between A(4,8) and B(1,12)
A (4, 8)
B (1, 12)
( x1 , y1 )and ( x2 , y2 )
distance  ( x2  x1 )  ( y2  y1 )
2
distance  (1  4)  (12  8)
2
distance  (3)  (4)
2
2
2
distance  9  16  25 
5
2
YOU TRY!!
 Find the distance between:
A. (2, 7) and (11, 9)
(9)  (2)  85
2
2
B. (-5, 8) and (2, - 4)
(7)  (12)  193
2
2
Midpoint Formula
Used to find the center of a line segment
( x1 , y1 )and ( x2 , y2 )
 x2  x1 y2  y1 
midpoint  
,

2 
 2
Example
Find the midpoint between A(4,8) and B(1,12)
A (4, 8)
B (1, 12)
 x2  x1 y2  y1 
midpoint  
,

2 
 2
 1  4 12  8 
midpoint  
,

2 
 2
midpoint   5 ,10 


2


YOU TRY!!
Find the midpoint between:
A) (2,
7) and (14, 9)
midpoint = 8,8
B) (-5,
8) and (2, - 4)
 -3 
midpoint =  , 2 
2 
THE SLOPE
FORMULA!

y2  y1
m
x2  x1
(6, 5)
x2 y2
Use the slope formula
y 2  y1
x2  x1
( -5, -3)
x1 y1
35
8
8
=  11 =
11
56
Homework

 Complete the handout given in class.
It is also posted on GradeSpeed and my website.
12.6
Midpoint and Distance
Formulas
Goal 1
Find the Midpoint of a
Segment
Goal 2
Two
Find the Distance Between
Points on a Coordinate Plane
The Midpoint Formula
The midpoint between the two points
(x1, y1) and (x2, y2) is:
x2  x1 y2  y1
m(
,
)
2
2
Example 1
Find the midpoint of the segment
whose endpoints are (6,-2) & (2,9)
Example 2
Find the coordinates of the
midpoint of the segment whose
endpoints are (5, 2) and ( 7, 8)
Example 3
Find the coordinates of the
midpoint of the segment whose
endpoints are (-2, 8) and ( 4, 0)
Distance Formula
The distance between two points with
coordinates (x1, y1) and (x2, y2) is given
by:
d  ( x2  x1 )  ( y2  y1 )
2
2
Example 4
Find the Distance Between the
points.
(-2, 5) and (3, -1)
Let (x1, y1) = (-2, 5) and (x2, y2) = (3, -1)
Example 5
What is the distance between P(- 1, 4) and
Q(2, - 3)?
Example 6
What is the distance between P(3, 0) and
Q(5, - 4)?
Example 7
What is the distance between P(-5, 2) and
Q(2, - 5)?
Use the distance formula to
determine whether the three points are
vertices of a right triangle: (1,1), (4,4), (4,1)
Example 8
Use the distance formula to
determine whether the three points are
vertices of a right triangle: (3, -4), (-2, -1),
(4, 6).
Example 9
p. 748 #16-28e, 36-44e, 61-63
There are formulas that you will be
provided with to calculate various
pieces of information about pairs of
points.
Each formula refers to a set of two
points:
(x1, y1) and (x2, y2)
Distance – the length of the line
segment that connects two given
points in the coordinate plane.
Distance Formula:
x2  x1 
2
 y2  y1 
2
Ex#1: (2, 2) and (5, -2)
Distance: ________
The midpoint is the point equidistant
between two points in the coordinate
plane.
Midpoint Formula:
 x 2  x1 y 2 y 1 
,


2 
 2
NOTICE: the answer to a midpoint formula problem
will be in the form (1, 2) – meaning your answer is
another point!
Ex# 1: (2, 2) and (5, -2)
Midpoint: _______
The slope is the ratio of vertical
change (rise) to horizontal change
(run) of a line.
Slope Formula:
y2  y1
m
x2  x1
Ex# 1: (2, 2) and (5, -2)
Slope: __________
Example 2: (0, 3) and (-1, 1)
Distance: ________
Midpoint: _______
Slope: __________
Slope: There are four classifications of slope:
positive, negative, zero, and undefined.
Positive
Skiing Uphill
Examples:
2 1
5
, , 2, 1,
3 2
2
Negative
Skiing Downhill
Examples:
1 3
 , ,3,8
3 5
Slope: There are four classifications of slope:
positive, negative, zero, and undefined.
Zero
Undefined
You have an
undefined slope
whenever you
get a zero in
the
denominator.
Cross Country Skiing
Examples:
0 0
0, ,
2 8
If you tried to ski on
this, you wouldn’t make it.
1 -3 8
Examples:
,
,
0 0 0
Independent Practice: Calculate the slope for
each pair of points. Classify each slope as
positive, negative, zero, or undefined.
1. (2, 2) and (3, 5)
2. (0, 0) and (3, 0)
Slope: 3
Classification: Positive
Slope: 0
Classification: Zero
3. (-2, -1) and (-1, -4)
4. (2, 3) and (2, 7)
Slope: -3 Classification:
Negative
Slope: Undefined
Classification: Undefined
5. (-1, -1) and (5, 5)
6. (8, 4) and (6, 4)
Slope: 1
Classification: Positive
Slope: 0
Classification: Zero
Lesson 1-3

Formulas
Lesson 1-3: Formulas
35
The Coordinate
Plane
plane,
the horizontal number line

Definition: In the coordinate
(called the x- axis) and the vertical number line
(called the y- axis) interest at their zero points
called the Origin.
y - axis
Origin
x - axis
Lesson 1-3: Formulas
36
The Distance Formula
The distance d between any two points with coordinates ( x1 , y1 )
2
and ( x 2 , y 2 ) is given by the formula d =
.
( x 2 - x1 ) + ( y 2 - y1 )
2
Example: Find the distance between (-3, 2) and (4, 1)
x1 = -3, x2 = 4, y1 = 2 , y2 = 1
d=
2
2
(-3 - 4) + (2 - 1)
2
2
d=
(-7) + (1) = 49 + 1
d=
50 or 5 2 or 7.07
Lesson 1-3: Formulas
37
Midpoint Formula
In the coordinate plane, the coordinates of the midpoint of a
segment whose endpoints have coordinates ( x1 , y1 ) and ( x2 , y2 )
are  x1  x2 , y1  y2  .

2
Example:
2

Find the midpoint between (-2, 5) and (6, 4)
x1 = -2, x2 = 6, y1 = 5, and y2 = 4
M=
æ -2 + 6 , 5 + 4 ö
è 2
2 ø
M=
æ 4 , 9 ö = æ 2, 9 ö
è 2 2ø
è 2ø
Lesson 1-3: Formulas
38
Slope Formula
Definition: In a coordinate plane, the slope of a line is the ratio of
its vertical rise over its horizontal run. rise
run
Formula: The slope m of a line containing two points with
coordinates ( x1 , y1 ) and ( x2 , y2 ) is given by
the formula
y2  y1
m
where x1  x2 .
x2  x1
Example: Find the slope between (-2, -1) and (4, 5).
x1  2, x2  4, y1  1, y2  5
m
y2  y1 5  (1)

x2  x1 4  (2)
Lesson 1-3: Formulas
6
m  1
6
39
Describing Lines
rise from left to right.
 Lines that have a positive slope
 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
40
Some More
Examples

Find the slope between (4, -5) and (3, -5) and describe it.
m=
-5 - -5 0
= =0
4 -3
1
Since the slope is zero, the line must be horizontal.

Find the slope between (3,4) and (3,-2) and describe the
line.
4 - -2 6
= =Æ
m=
3- 3
0
Since the slope is undefined, the line must be vertical.
Lesson 1-3: Formulas
41
Example 3 : Find the slope of the line through the
given points and describe the line.
(7, 6) and (– 4, 6)
y
Solution:
y2  y1
m 
up 0
x2  x1
66

(4)  7
0

11
0
This line is horizontal.
(– 4, 6)
left 11
(-11)
(7, 6)
x
Lesson 1-3: Formulas
42
Example 4: Find the slope of the line through the
given points and describe the line.
right 0
(– 3, – 2) and (– 3, 8)
(– 3, 8)
Solution:
m

y
y2  y1
x2  x1
8   2 

(3)   3
10

0
up 10
x
(– 3, – 2)
undefined
This line is vertical.
Lesson 1-3: Formulas
43
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.
Lesson 1-3: Formulas
44
Example 1
Use the Distance Formula
Example 2
Use the Distance Formula to Solve a Problem
Example 3
Use the Midpoint Formula
Find the distance between M(8,
4) and N(–6, –2). Round to the
nearest tenth, if necessary.
Use the Distance Formula.
Distance Formula
Simplify.
Evaluate (–14)2
and (–6)2.
Add
196 and 36.
Take the square root.
Answer:
The distance between points M and N is
about 15.2 units.
Find the distance between A(–4,
5) and B(3, –9). Round to the
nearest tenth, if necessary.
Answer:
The distance between points A and B is
about 15.7 units.
Geometry Find the perimeter of XYZ to the
nearest tenth.
First, use the Distance Formula to find
the length
of each side of the triangle.
Distance Formula
Simplify.
Evaluate powers.
Simplify.
Distance Formula
Simplify.
Evaluate powers.
Simplify.
Distance Formula
Simplify.
Evaluate powers.
Simplify.
Then add the lengths of the sides to find the perimeter.
Answer:
The perimeter is about 15.8 units.
Geometry Find the perimeter of ABC to the
nearest tenth.
Answer:
The perimeter is
about 21.3 units.
Find the coordinates of the midpoint of
Midpoint Formula
Substitution
Simplify.
Answer:
The coordinates of the midpoint of
are (3, 3).
Find the coordinates of the midpoint of
Answer:
The coordinates
of the midpoint
of
(1, –1).
are