1-1b: The Coordinate Plane - Distance Formula

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1-1b The Coordinate Plane- Distance Formula
Pythagorean Theorem
CCSS
G-CO.1 Know precise definitions of angle, circle,
perpendicular line, parallel line, and line
segment, based on the undefined notions of point,
line, distance along a line, and distance around
a circular arc.
GSE
M(GM)109 Solves problems on and off the
coordinate plane involving distance, midpoint,
perpendicular and parallel lines, or slope
M(GM)102 Makes and defends conjectures,
constructs geometric arguments, uses geometric
properties, or across disciplines or contexts
(e.g., Pythagorean Theorem
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Example Find the measure of AB.
A
B
Point A is at 1.5 and B is at 5. So, AB 5 -
1.5 3.5
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Example

• Find the measure of PR
• Ans 3-(-4)347
• Would it matter if I asked for the distance from
  R to P ?

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Ways to find the length of a segment on the
coordinate plane

• 1) Pythagorean Theorem- Can be used on and off
  the coordinate plane

• 2) Distance Formula only used on the coordinate
  plane

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1) Pythagorean Theorem

• Only can be used with Right Triangles
• What are the parts to a RIGHT Triangle?
• Right angle
• 2 legs
• Hypotenuse

Hypotenuse- Side across from the right angle.
Always the longest side of a right triangle.
LEG
Right angle
Leg Sides attached to the Right angle
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Pythagorean Formula
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Example of Pyth. Th. on the Coordinate Plane
Make a right Triangle out of the segment
(either way)
Find the length of each leg of the right Triangle.
Then use the Pythagorean Theorem to find the
Original segment JT (the hypotenuse).
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Find the length of CD using the Pythagorean
Theorem
We got 10 by 6 - - 4
10
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We got 8 by -4 4
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Ex. Pythagorean Theorem off the Coordinate Plane

• Find the missing segment- Identify the parts of
  the triangle

Leg
5 in
Leg 2 Leg 2 Hyp 2
Ans 5 2 X 2 13 2
Leg
13 in
25 X 2 169
hyp
X 2 144
X 12 in
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2) Distance Formula
Lets Use the Pythagorean Theorem
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d
J (-3,5) T (4,2)
x1, y1 x2, y2
Identify one as the 1st point and one as the 2nd.
Use the corresponding x and y values
(4-(-3))2 (2-(5))2
(43)2 (2-5)2
(7)2 (-3)2
58
499


7.6

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Example of the Distance Formula

• Find the length of
• the green segment
• Ans 109 or approximately 10.44

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( ) Congruent Segments

• Segments that have the same length.
• If AB XY have the same length,
• Then ABXY,
• but
• AB XY

Symbol for congruent
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Assignment