The Coordinate Plane

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The Coordinate Plane

• During this lesson you will
• Find the distance between two points in the
  plane
• Find the coordinates of the midpoint of a
  segment

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PART I FINDING DISTANCE
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The Coordinate Plane
Quadrant I (, )
Quadrant II (-, )
T
The coordinates of point T are ________.
(6,3)
(0,0)
Origin ?
Quadrant III (-, -)
Quadrant IV (, -)
The Coordinate Plane
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When working with Coordinate Geometry, there are
many ways to find distances (lengths) of line
segments on graph paper. Let's examine some of
the possibilities
Method 1 Whenever the segments are horizontal or
vertical, the length can be obtained by
counting.  
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Method One
Unfortunately, this counting approach does NOT
work for EF which is a diagonal segment.

• Whenever the segments are horizontal or
  vertical, the length can be obtained by counting.
  
• When we need to find the length (distance) of
  a segment such as AB, we simply COUNT the
  distance from point A to point B.(AB ___)
• We can use this same counting approach for CD
  .(CD ___)

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3
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Method 2 To find the distance between two
points, A(x1, y1) and B(x2, y2), that are not on
a horizontal or vertical line, we can use the
Distance Formula.
Formula The Distance Formula
The distance, d, between two points, A(x1, y1) and B(x2, y2), is The distance, d, between two points, A(x1, y1) and B(x2, y2), is
Alert! The Distance Formula can be used for all
line segments vertical, horizontal, and
diagonal.
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Finding Distance
ALERT! Order is important when using Distance
Formula.

• What is the distance between the two points on
  the right?
• STEP 1 Find the coordinates of the two
  points.____________
• STEP 2 Substitute into the Distance Formula.

(6,8)
(0,0)
(6,8)
(0,0)
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Example Given (0,0) and (6,8), find the
distance between the two points.
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Applying the Distance Formula
Each morning H. I. Achiever takes the bus line
from Oak to Symphony. How far is the bus ride
from Oak to Symphony?
(2,4)Jackson
(__,__) North
(__,__) Central
(__,__) Symphony
(__,__) City Plaza
(__,__) Cedar
(__,__) Oak
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Final Checks for Understanding

1. State the Distance Formula in words.
2. When should the Distance Formula be used when
   determining the distance between two given
   points?
3. Find the length of segment AB given A (-1,-2)
   and B (2,4).

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Homework Assignment

• Page 46, text 1-17 odd.
• Extra Practice WS Distance Formula with
  Solutions Available Online

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PART II FINDING THE MIDPOINT OF A SEGMENT
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Vocabulary

• midpoint of a segment - _________________________
  _________________________________________________
  _________

point
on a segment which divides the segment into two
congruent segments
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In Coordinate Geometry, there are several ways
to determine the midpoint of a line segment.
Method 1 If the line segments are vertical or
horizontal, you may find the midpoint by simply
dividing the length of the segment by 2 and
counting that value from either of the
endpoints.  
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Method 1 Horizontal or Vertical Lines

• If the line segments are vertical or
  horizontal, you may find the midpoint by simply
  dividing the length of the segment by 2 and
  counting that value from either of the endpoints. 

 
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The Midpoint Formula works for all line
segments  vertical, horizontal or diagonal.
To find the coordinates of the midpoint of a
segment when the lines are diagonal, we need to
find the average (mean) of the coordinates of the
midpoint.
The Midpoint Formula The midpoint of a segment
endpoints (x1 , y1) and (x2 , y2) has
coordinates                                     
                              
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Finding the Midpoint

• Find the midpoint of line segment AB.
• A (-3,4)
• B (2,1)
• Check your answer here

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Consider this tricky midpoint problem

• M is the midpoint of segment CD.  The
  coordinates M(-1,1) and C(1,-3) are given.  Find
  the coordinates of point D.

First, visualize the situation.  This will give
you an idea of approximately where point D will
be located.  When you find your answer, be sure
it matches with your visualization of where the
point should be located.
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Solve algebraicallyM(-1,1), C(1,-3) and
D(x,y)Substitute into the Midpoint Formula
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Solve for each variable separately
(-3,5)
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Other Methods of Solution
Verbalizing the algebraic solution Some
students like to do these "tricky" problems by
just examining the coordinates and asking
themselves the following questions"My
midpoint's x-coordinate is -1.  What is -1 half
of? (Answer -2)What do I add to my endpoint's
x-coordinate of 1 to get -2? (Answer -3)This
answer must be the x-coordinate of the other
endpoint."These students are simply verbalizing
the algebraic solution.  (They use the same
process for the y-coordinate.)
 
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Final Checks for Understanding

1. Name two ways to find the midpoint of a given
   segment.
2. What method for finding the midpoint of a segment
   works for all lineshorizontal, vertical, and
   diagonal?
3. Explain how to find the coordinates of an
   endpoint when you are given an endpoint and the
   midpoint of a segment.

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Homework Assignment
Page 46, text 1-17 odd. Extra Practice WS
Midpoint Formula with Solutions Available Online
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Solution

• Given A(-3,4) B(2,1)
• The midpoint will have
• coordinates
• Alert! Your answer may contain a fraction. 
  Answers may be written in fractional or decimal
  form.

Answer
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