Midpoints

1
Midpoints
2
What is the midpoint?

4
0
3
How did you get it?

4
0
2
4/2 2
4
What is the midpoint?

5
1
5
How did you get it?

5
1
(15)/2 3
6
In other words,its just the average!
5
3
3
6
1
7
In two dimensions
(8, 6)
(2, 2)
8
Average the x-coords
2
8
9

2
8
10
(5, _ )

2
8
(28)/2
11
Average the y-coords
6
(5, _ )
2
12
6

(5, _ )
2
13
6

(26)/2
(5, 4 )
2
14
The Midpoint (5,4)
(8, 6)
(26)/2

(5, 4)
(2, 2)

(28)/2
15
(No Transcript)
16
The Algorithm Method
2. ( 2 , 2 ) 2nd Point
3. 10 8 sums
4. ( 5 , 4 ) halves
17
Find the midpoint between
(6, 0) and (2, 4)
(4, 2)
18
Find the midpoint between
(2, -3) and (4, 2)
(3, -½)
19
Find the midpoint between
(-4, -3) and (2, 2)
(-1, -½)
20
The Distance Formula

• But first

21
The Pythagorean Theorem
5
3
4
22
The Pythagorean Theorem
9
5
3
4
23
The Pythagorean Theorem
9
5
3
4
16
24
The Pythagorean Theorem
25
9
5
3
4
16
25
The Pythagorean Theorem
25
9
5
3
25 9 16
4
16
26
The Pythagorean Theorem
25
9
5
3
52 32 42
4
16
27
The Pythagorean Theorem
c
a
c2 a2 b2
b
28
The Pythagorean Theorem
c
a
b
The length of the hypotenuse of a right triangle
is equal to the square root of the sum of the
squares of the length of the legs.
29
Finding distance
(6, 5)
(2, 2)
30
using the right triangle.
(6, 5)
5
2
(2, 2)
6
2
31
We need a and b to get c.
5
c
b
2
a
6
2
32
The Difference of the Coordinates
5
c
b5-2
2
a6-2
6
2
33
The Difference of the Coordinates
5
c
b5-2
2
a6-2
6
2
34
c
3
4
35
Finding distance
(x2, y2)
(x1, y1)
36
using the right triangle.
(x2, y2)
y2
y1
(x1, y1)
x2
x1
37
The Difference of the Coordinates
y2
c
b y2- y1
y1
a x2-x1
x2
x1
38
c v(x2-x1)2 (y2- y1 ) 2
c
y2- y1
x2-x1
39
Does order matter? Why?
distance v(x2-x1)2 (y2- y1 ) 2
distance v(x1-x2)2 (y1- y2 ) 2
(5-3)2 (2)2 4
(3-5)2 (-2)2 4
40
The Algorithm Method
2. - ( 2 , 2 ) 2nd Point
3. 4 3 differences
4. 16 9 squares
5. 25 sum
6. 5 sq. root
41
Find the distance between
(2, -3) and (4, 2)
v29
42
Find the distance between
(-4, -3) and (-6, 2)
5v5
43
Find the distance between
(6, 0) and (2, 4)
4v2
44
Homework

• Page 163
• 15-29, 39-48