Transcript The Coordinate Plane

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
Mrs. McConaughy
Geometry
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PART I: FINDING
DISTANCE
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Geometry
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The
Coordinate
Quadrant
II (-, +) Quadrant IPlane
(+, +)
T
Origin 
Quadrant III (-, -)
(0,0)
The coordinates of point
T are (6,3)
________.
Quadrant IV (+, -)
The Coordinate Plane
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Geometry
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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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Unfortunately, this counting approach does NOT
Method One
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.
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(AB = ___)
We can use this same
counting approach for CD .
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(CD = ___)
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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
Alert! The Distance Formula can be used for
all line segments: vertical, horizontal, and
diagonal.
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ALERT! Order is important when using
Finding
Distance
Distance Formula.
What is the distance
between the two
points on the right?
STEP 1: Find the
coordinates of the two
(0,0)
(6,8)
points.____________
STEP 2: Substitute into
the Distance Formula.
(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?
Mrs. McConaughy
(2,4)Jackson
(__,__) North
(__,__) Central
(__,__) Cedar
(__,__) Symphony
(__,__) City Plaza
(__,__) Oak
Geometry
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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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Geometry
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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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Geometry
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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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Geometry
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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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Geometry
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To find the coordinates of the
The Midpoint Formula
works
midpoint of a segment when the
for all linelines
segments:
are diagonal, we need to find
theor
average
(mean) of the
ertical, horizontal
diagonal.
coordinates of the midpoint.
The Midpoint Formula:
The midpoint of a segment
endpoints (x1 , y1) and (x2 , y2)
has coordinates
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Geometry
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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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Geometry
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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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Geometry
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Solve algebraically:
M(-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.
Mrs.(They
McConaughy
use the same process Geometry
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 lines…horizontal,
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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Geometry
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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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Geometry
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Solution
Given: A(-3,4); B(2,1)
The midpoint will have
coordinates:
Answer:
Alert!
Your answer may contain a
fraction. Answers may be written in
fractional or decimal form.
Click here to return to lesson.
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Geometry
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