Solving the Distance, Midpoint, and Slope.ppt

Solving the Distance, Midpoint, and Slope

1. Coordinate Geometry and its Formulas
(Distance,Midpoint,Slope)

2. We will talk about 3 formulas that
are used to calculate various pieces
of information about pairs of points.
Each formula refers to a set of two
points:
(x
1
, y
1
) and (x
2
, y
2
)

3. Distance between two points.
5 18
3
17
A(5,3)
B(18,17)
18 – 5 = 13 units
17 – 3 = 14 units
AB
2
= 13
2
+ 14
2
Using Pythagoras’
Theorem,
AB
2
= (18 - 5)
2
+ (17 - 3)
2
y
x

4. Distance between two points.
In general,
x
1
x
2
y
1
y
2
A(x
1
,y
1
)
B(x
2
,y
2
)
Length = x
2 – x
1
Length = y
2
– y
1
AB
2
= (y
2-y
1)
2
+ (x
2-x
1)
2
Hence, the formula for
Length of AB or Distance
between A and B is
y
x

5. The Distance Formula
2 2
2 1 2 1
( ) ( )d x x y y   
"The Distance Formula"
sung to the tune of "On Top of Old Smokey"
When finding the distance
Between the two points,
Subtract the two x's
The same for the y's.
Now square these two numbers,
And find out their sum.
When you take the square root
Then you are all done!

6. Ex#1: (2, 2) and (5, -2)
Distance: ________

8. Midpoint Formula:





 
2
,
2
1212
yyxx

9. The Midpoint Formula
1 2 1 2
( , ) ,
2 2
x x y y
xy
  

 
 
"The Midpoint Formula"
sung to the tune of "The Itsy Bitsy Spider"
When finding the midpoint of
two points on a graph,
Add the two x's and cut their
sum in half.
Add up the y's and divide 'em by
a two,
Now write 'em as an ordered
pair
It’s the middle of the two.

10. Ex# 1: (2, 2) and (5, -2)
Midpoint: _______

11. ExM(4, 2) is the midpoint of RS. If S has a coordinates (5, -2),
find the coordinates of R.
2, 2
( )x y

2


5
1
x
1
5x  8
( , )3

2


2 1
y
6
,
( )
m m
x y1, 1
( )x y
R (x
1
, y
1
)
S (5,-2)
M(4, 2)
4
2
(2) (2)
5 5
1
x3
(2) (2)
1
2y  4
2 2
1
y6

12. “Real”-world example
On a road trip, you hike 3 miles north and 2miles
west. Starting at the same point, your friend
hikes 4 miles east and 1 mile south.
How far apart are you?
If you want to meet for lunch, where could you
meet so each person goes the same distance?

13. Find the distance between a point
and a line.
What is the distance from point B to line q?
B
q
(1, 4)
(4, 1)
(1, -2)

14. Example
Find the distance from point A to line c.

15. The slope is the ratio of vertical
change (rise) to horizontal change
(run) of a line.
Slope Formula:
12
12
xx
yy
m




16. Ex: (2, 2) and (5, -2)
Slope: __________

17. (0, 3) and (-1, 1)
Distance: ________
Midpoint: _______
Slope: __________