Transcript Geometry Notes - Eastern Upper Peninsula ISD

Geometry Notes
Section 1-3
9/7/07
What you’ll learn


How to find the distance between two points
given the coordinates of the endpoints.
How to find the coordinate of the midpoint of a
segment given the coordinates of the endpoints.
 How to find the coordinates of an endpoint
given the coordinates of the other endpoint
and the midpoint.
Vocabulary Terms:
Midpoint
 Segment bisector

Midpoint
In general the midpoint is the exact middle
point in a line segment, but how do we
define it geometrically?
 If M is going to be the midpoint of PQ, then
what rules does it have to follow?

Q
P
M
Geometric definition of a segment’s
midpoint. . .
Q
P
M
Does the midpoint have to be located
anywhere special?
 YUP
 Between the endpoints P and Q.
 Rule #1: M must be between P and Q.

 Remember
this implies collinearity
 And PM + MQ = PQ
Any other requirements for midpoint?
Q
 Yup—
P
M
 It
has to cut the segment in half. How do
we express that geometrically?
In
half means in two equal pieces. . .
Equal pieces—Equal length or CONGRUENT
 Rule
PM
#2:
= MQ or PM  MQ.
Can you identify and model a
segment’s midpoint? Q

M
P
How do you model/illustrate equal length
or congruence?
 Identical markings on congruent
parts/pieces.

Now to find the length of the
segment or distance between the
endpoints. . . .
 First
consider a simple number line.
 Then
we’ll look at the coordinate
plane.
Finding the distance between 2 pts
on a number line.

Use the coordinates of a line segment
to find its length.

Consider a simple number line:
P
Q
-3 -2 -1 0 1 2 3 4 5 6

How would you find PQ?
To find the distance between two
points on a number line:

Subtract the coordinates then take the
absolute of that number (remember
distance can’t be negative).
One dimensional – piece of cake. .
4
3
2
What happens with 2-dimensions?
1
-6
-4
-2
2
-1
2-Dimensional refers to a
coordinate plane
-2
-3
-4
4
6
How to find distance on a
coordinate plane
 There
are two methods
Pythagorean theorem
Distance Formula
Everyone knows the Pythagorean
theorem. . . .
 a2
+ b 2 = c2
Where a, b, and c refer to the sides
of a RIGHT triangle. . .
How do we get a right triangle out
of a line segment?
44
33
AB = 5
22
11
-6
-6
-4
-4
-2
-2
A
B
a=4
22
b=3
-2
-2
-3
-3
-4
-4
66
a2 + b2 = c2
 42 + 32 = (AB)2
 16 + 9 = (AB)2

25 = (AB)2

5 = AB

-1
-1
44
In order to use the Pythagorean
theorem. . . .
 You
have to complete the right
triangle.
What if the numbers are too big to
graph?
 There
has to be another way. . .
The Distance Formula

The distance between two points with
coordinates (x1, y1) and (x2, y2)
distance =

( x2  x1 )  ( y2  y1 )
2
Using the same segment in our earlier
example. . . .
2
The distance between two points with
coordinates A(-2, -1) and B(1, 3)
4
AB =
( x2  x1 )  ( y2  y1 )
2


AB =
(1 2)  ( 3 1)
AB =
( 3)  ( 4 )
AB =
9 +16
AB =
AB = 5
2
2
-6
25
-4
2
2
2
3
B: (1.00, 3.00)
B
2
1
-2
2
A: (-2.00, -1.00)
A
Look
familiar???
-1
-2
-3
There is a relationship between the
Pythagorean Theorem and the Distance
Formula. . . .

If you solve a2 + b2 = c2 for c, you will get
c a b
2

2
a and b represent the vertical and
horizontal distances from the right triangle
 vertical
distance = subtracting the ycoordinates
2
1
 horizontal distance = subtracting the xcoordinates
2
1
a  (x  x )
b (y  y )

The distance formula related to the
Pythagorean theorem because. . .
a  ( x2  x1 )
c a b
2
2
b  ( y2  y1 )
So. . . .
distance = ( x2  x1 )  ( y2  y1 )
2
2
Can you find distance on a
coordinate plane?
 Using
both
methods?
Pythagorean
a2 + b2 = c2
theorem
Distance
2
2
distance = ( x2  x1 )  ( y2  y1 )
Formula
Finding the location (coordinate) of the
midpoint
On a number line. . . .
 Recall the midpoint is exactly half way
between the endpoints of a segment

P
Q
-3 -2 -1 0 1 2 3 4 5 6
At what coordinate is the midpoint of PQ
located?
 The midpoint would be located at 2.5

Finding the location (coordinate) of the
midpoint mathematically


On a number line. . . .
The coordinate of the midpoint is the average of
the coordinates of the endpoints
P
Q
-3 -2 -1 0 1 2 3 4 5 6

HUH?
Average the coordinates of the
endpoints. . . .

Formula:
a is the
coordinate of
one endpoint
b is the
coordinate of the
other endpoint
ab
midpt 
2
Back to our example. . . .
P
Q
-3 -2 -1 0 1 2 3 4 5 6
 Formula:
 1 is the
midpt
coordinate of
one endpoint
 4 is the
midpt
coordinate of the
other endpoint
1 4

2
 2.5
Finding the location (coordinate) of the
midpoint on a coordinate plane
Basically it’s the same as finding the
midpoint on a number line
 Recall the midpoint is exactly half way
between the endpoints of a segment
 We averaged the coordinates for a
number line and we will average the
coordinates for a coordinate plane

Average the coordinates of the
endpoints. . . .

Formula:
(x1, y1) is the coordinate of one endpoint
(x2, y2) is the coordinate of the other
endpoint
 x1  x2 y1  y2 
 xm , y m   
,

2 
 2
Find the coordinate of the midpoint
of AB.
4
3
B: (1.00, 3.00)
B
2
1
-6
-4
-2
2
A: (-2.00, -1.00)
A
-1
-2
-3
-4
4
6
We know: A(-2, -1) B(1, 3)
 x1  x2 y1  y2 
Formula: xm , y m   
,

2 
 2
  2  1 1  3 
,

Fill It In:  xm , ym   
2 
 2
Simplify It:
 1 
xm , ym    ,2 
 2 
Find the coordinate of the missing
endpoint…
4
C
3
2
1
-6
-4
M: (1.00, 1.00)
M
-2
2
D: (-2.00, -1.00)
D
-1
-2
-3
-4
4
6
We know
(xm, ym) is (1, 1) and (x1, y1) is (-2, -1)
 x1  x2 y1  y2 


x
,
y

,


m
m
Formula:
2 
 2
  2  x2  1  y2 
Fill It In:  xm , ym   
,

2 
 2
Split It:
2  x 2
1
2
1  y 2
1
2
Solve for x2:
1 2  x 2

1
2
2  x 2
1
2
1( 2  x2 )  1( 2)
2  x2  2
x2  4
1  y 2
1
2
1 1  y 2

1
2
1( 1  y2 )  1( 2)
Solve for y2:
1  y2  2
y2  3
FINALLY our answer is . . . .
(4, 3)
Have you learned. . .
How to find the distance between two
points given the coordinates of its
endpoints?
 How to find the coordinate(s) of the
midpoint of a segment given the
coordinates of the endpoints?

 How
to find the coordinates of an endpoint
given the coordinates of the other endpoint
and the midpoint?
Assignment: Worksheet 1.3