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