Geometry Notes

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Geometry Notes

• Section 1-3
• 9/7/07

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What youll 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.

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Vocabulary Terms

• Midpoint
• Segment bisector

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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?

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Geometric definition of a segments 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 collinearity
• And PM MQ PQ

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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 piecesEqual length or CONGRUENT
• Rule 2
• PM MQ or PM ? MQ.

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Can you identify and model a segments midpoint?

• How do you model/illustrate equal length or
  congruence?
• Identical markings on congruent parts/pieces.

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Now to find the length of the segment or distance
between the endpoints. . . .

• First consider a simple number line.
• Then well look at the coordinate plane.

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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?

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To find the distance between two points on a
number line

• Subtract the coordinates then take the absolute
  of that number (remember distance cant be
  negative).

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One dimensional piece of cake. .
What happens with 2-dimensions?
2-Dimensional refers to a coordinate plane
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How to find distance on a coordinate plane

• There are two methods
• Pythagorean theorem
• Distance Formula

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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?

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AB 5
a 4

• a2 b2 c2
• 42 32 (AB)2
• 16 9 (AB)2
• 25 (AB)2
• 5 AB

b 3
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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. . .

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The Distance Formula

• The distance between two points with coordinates
  (x1, y1) and (x2, y2)

• Using the same segment in our earlier example. .
  . .

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The distance between two points with coordinates
A(-2, -1) and B(1, 3)
Look familiar???
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There is a relationship between the Pythagorean
Theorem and the Distance Formula. . . .

• If you solve a2 b2 c2 for c, you will get

• a and b represent the vertical and horizontal
  distances from the right triangle
• vertical distance subtracting the y-coordinates
  
• horizontal distance subtracting the
  x-coordinates

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So. . . .

• The distance formula related to the Pythagorean
  theorem because. . .

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Can you find distance on a coordinate plane?

• Using both methods?
• Pythagorean theorem
• Distance Formula

a2 b2 c2
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Finding the location (coordinate) of the midpoint

• On a number line. . . .
• Recall the midpoint is exactly half way between
  the endpoints of a segment

• At what coordinate is the midpoint of PQ located?
• The midpoint would be located at 2.5

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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

• HUH?

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Average the coordinates of the endpoints. . . .

• Formula
• a is the coordinate of one endpoint
• b is the coordinate of the other endpoint

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Back to our example. . . .

• Formula
• 1 is the coordinate of one endpoint
• 4 is the coordinate of the other endpoint

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Finding the location (coordinate) of the midpoint
on a coordinate plane

• Basically its 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

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Average the coordinates of the endpoints. . . .

• Formula
• (x1, y1) is the coordinate of one endpoint
• (x2, y2) is the coordinate of the other endpoint

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Find the coordinate of the midpoint of AB.
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We know A(-2, -1) B(1, 3)
Formula
Fill It In
Simplify It
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Find the coordinate of the missing endpoint
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We know (xm, ym) is (1, 1) and (x1, y1) is (-2,
-1)
Formula
Fill It In
Split It
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Solve for x2
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Solve for y2
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FINALLY our answer is . . . .

• (4, 3)

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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