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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. - PowerPoint PPT PresentationTRANSCRIPT
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?
P
QM
Geometric definition of a segment’s midpoint. . .
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 collinearityAnd PM + MQ = PQ
P
QM
Any other requirements for midpoint?
Yup— 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 #2:PM = MQ or PM MQ.
P
QM
Can you identify and model a segment’s midpoint? M
P
Q
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.
P Q
Consider a simple number line:Consider a simple number line:
-3 -2 -1 0 1 2 3 4 5 6
How would you find PQ?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. .
What happens with 2-dimensions?
2-Dimensional refers to a coordinate plane
4
3
2
1
-1
-2
-3
-4
-6 -4 -2 2 4 6
How to find distance on a coordinate plane
There are two methodsPythagorean theoremDistance Formula
Everyone knows the Pythagorean theorem. . . .
a2 + b2 = 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?
4
3
2
1
-1
-2
-3
-4
-6 -4 -2 2 4 6
A
B
4
3
2
1
-1
-2
-3
-4
-6 -4 -2 2 4 6
A
B
4
3
2
1
-1
-2
-3
-4
-6 -4 -2 2 4 6
A
B
a = 4
b = 3
a2 + b2 = c2
42 + 32 = (AB)2
16 + 9 = (AB)2
25 = (AB)2
5 = AB
AB = 5
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 = ( ) ( )x x y y2 12
2 12
Using the same segment in our earlier example. . . .
The distance between two points with coordinates A(-2, -1) and B(1, 3)
AB = ( ) ( )x x y y2 12
2 12
AB = ( ) ( )1 2 3 12 2
AB = ( ) ( )3 42 2
4
3
2
1
-1
-2
-3
-4
-6 -4 -2 2 4 6
A: (-2.00, -1.00)
B: (1.00, 3.00)
A
B
Look familiar???
AB = 9 +16
AB = 25
AB = 5
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 y-coordinates horizontal distance = subtracting the x-
coordinatesa x x ( )2 1
b y y ( )2 1
So. . . .
The distance formula related to the Pythagorean theorem because. . .
c a b 2 2
distance = ( ) ( )x x y y2 12
2 12
a x x ( )2 1
b y y ( )2 1
Can you find distance on a coordinate plane?
Using both methods?Pythagorean
theoremDistance
Formuladistance = ( ) ( )x x y y2 1
22 1
2
a2 + b2 = c2
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
midpta b
2
Back to our example. . . .
Formula: 1 is the
coordinate of one endpoint
4 is the coordinate of the other endpoint
midpt 1 42
P Q
-3 -2 -1 0 1 2 3 4 5 6
midpt 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
2
,2
, 2121 yyxxyx mm
Find the coordinate of the midpoint of AB.
4
3
2
1
-1
-2
-3
-4
-6 -4 -2 2 4 6
A: (-2.00, -1.00)
B: (1.00, 3.00)
A
B
We know: A(-2, -1) B(1, 3)
Formula:
Fill It In:
Simplify It:
2
,2
, 2121 yyxxyx mm
2
31,
2
12, mm yx
2,2
1, mm yx
Find the coordinate of the missing endpoint…
4
3
2
1
-1
-2
-3
-4
-6 -4 -2 2 4 6
M: (1.00, 1.00)
D: (-2.00, -1.00)
M
D
C
We know (xm, ym) is (1, 1) and (x1, y1) is (-2, -1)
12
22 x
11
22 y
2
,2
, 2121 yyxxyx mmFormula:
Fill It In:
Split It:
2
1,
2
2, 22 yxyx mm
Solve for x2:11
22
2 x
12
22 x
1 2 1 22( ) ( ) x
x2 4
2 22x
Solve for y2:1
12
2 y
11
12
2 y
1 1 1 22( ) ( ) y
1 22y
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