TECHNICAL GEOMETRY 4'4 The Pythagorean Theorem and the Distance Formula

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TECHNICAL GEOMETRY4.4 The Pythagorean Theorem
and the Distance Formula
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How to name the parts on a right triangle

• Legs are the sides that form the right angle
• Hypotenuse is the side opposite the right angle

B
hypotenuse
leg
A
leg
C
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How to name the parts on a right triangle
B

• We should label each
• side using the letter of
• the angle opposite it
• in lower-case (small letters)

c
a
A
b
C
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The Pythagorean Theorem

• In a right triangle, the square of the length of
  the hypotenuse is equal to the sum of the squares
  of the lengths of the legs.
• (hypotenuse)2 (leg)2 (leg)2

B
c
a
A
b
C
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The Pythagorean Theorem(named after Pythagoras)

• Pythagoras was a Greek philosopher who made
  important developments in mathematics, astronomy,
  and the theory of music. The theorem now known as
  the Pythagorean theorem was known to the
  Babylonians 1000 years earlier but he may have
  been the first to prove it.

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Using the Pythagorean Theorem how to calculate
the length of the hypotenuse

• (hypotenuse)2 (leg)2 (leg)2

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Using the Pythagorean Theorem how to calculate
the length of each leg

• (hypotenuse)2 (leg)2 (leg)2

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Using the Pythagorean Theorem how to calculate
the length of each leg

• (hypotenuse)2 (leg)2 (leg)2

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Try this!
Find the length of the hypotenuse
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Try this!
Find the unknown side length
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Using the Pythagorean Theorem Pythagorean
triples

• A Pythagorean triple is a set of
• three positive integers a, b, c
• that satisfy the equation

B
c
a
A
b
C
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Using the Pythagorean Theorem Solving
Pythagorean triples

• The three integers 3, 4, and 5
• form a Pythagorean triple, since
• 52 32 42

B
c
a
A
b
C
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Try this!

• Let c 13 and b 12,
• What is the value of a,
• so that a, b, and c form
• a Pythagorean triple.

Answer a 5
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Using Pythagorean Triples
Use the triples to find c Look at the sides
given (6, 8, c) Look for a common factor
between the numbers 2(3, 4, x) What triple is
in the pattern? (3,4,5) so 2(3,4,5) would be my
triangle so c is 2 times 5 so c 10
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Using Pythagorean Triples
Use the triples to find c Look at the sides
given (24, b, 51) Look for a common factor
between the numbers 3(8, x, 17) What triple is
in the pattern? (8,15,17) so 3(8,15,17) would
be my triangle so b is 3 times 15 so b 45
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Using the Pythagorean Theorem to find the length
of a segment on the coordinate plane.

• Find the length of segment AB,
• A(1,2) and B(4,6)

From counting a 4 and b 3 so c2 42
32 c2 16 9 c2 25 c 5
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The Distance Formula

• If A(x1,y1) and B(x2,y2)
• are points in a coordinate
• plane, then the distance
• between A and B is

(x2,y2)
y2 - y1
(x2,y1)
(x1,y1)
x2 - x1
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Using the Distance Formula

• Find the distance between
• A(-8,-6) and B(7,8)