NGSSS MA.8.G.2.4 The student will be able to:

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NGSSSMA.8.G.2.4The student will be able to
Validate and apply Pythagorean Theorem to find
distances in real world situations or between
points in the coordinate plane.
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CCSS8.G Understand and apply the Pythagorean
Theorem.
6. Explain a proof of the Pythagorean Theorem
and its converse.   7. Apply the Pythagorean
Theorem to determine unknown side lengths in
right triangles in real-world and mathematical
problems in two and three dimensions.   8. Apply
the Pythagorean Theorem to find the distance
between two points in a coordinate system.
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The Pythagorean Theorem in HS

• The Pythagorean Theorem underlies several
  formulas and identities that are memorized
• by high school students. Related formulas include
  
• The Distance formula
• The Law of Cosines
• The equation of a Circle
• Some trigonometric identities.
• Often, students memorize these formulas in
  isolation, without being aware of their
  connection to the Pythagorean Theorem.

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What is a right triangle?
hypotenuse
leg
right angle
leg

• It is a triangle which has an angle that is 90
  degrees.
• The two sides that make up the right angle are
  called legs.
• The side opposite the right angle is the
  hypotenuse.

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The Pythagorean Theorem

• In a right triangle, if a and b are the measures
  of the legs and c is the hypotenuse, then
• a2 b2 c2.
• Note The hypotenuse, c, is always the longest
  side.

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a2 b2 c2
http//www.pbs.org/wgbh/nova/proof/puzzle/theorem.
html
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Proof of the Pythagorean Theorem

• The image is the logo
• from the Institute for
• Mathematics Education.
• It provides us with an
• elegant geometric proof
• of the Pythagorean
• Theorem.
• Activity How does this illustration prove the
  Pythagorean Theorem?

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Proof of the Pythagorean Theorem

• Given the red right
• triangle, prove that the
• area of the square of the
• hypotenuse is equal to
• the sum of the areas of
• the squares of the two
• legs.
• The figure is formed from two large adjacent
  squares.
• Each large square contains four congruent right
  triangles, one of which is colored red.

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Proof of the Pythagorean Theorem

• The left square contains
• two smaller squares.
• The smallest square is
• the result of the shorter
• leg of the red right triangle.
• The larger square is the result of the longer leg
  of the red right triangle.
• The largest square at the right is the result of
  the hypotenuse of the red triangle.

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Proof of the Pythagorean Theorem

• Since both large squares
• are equal, we can
• subtract the four right
• triangles from each
• large square and still
• have equal areas.
• On the left are the squares of the two legs of
  the red right triangle. On the right is the
  square of the hypotenuse.
• Therefore, in a right triangle, the sum of the
  squares of the two legs is equal to the square
  of the hypotenuse.

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1. Find the length of the hypotenuse

• 122 162 c2
• 144 256 c2
• 400 c2
• Take the square root
• of both sides.

12 in
16 in
The hypotenuse is 20 inches long.
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2. Find the length of the hypotenuse

• 52 72 c2
• 25 49 c2
• 74 c2
• Take the square root of both sides.

5 cm
7 cm
The hypotenuse is about 8.6 cm long.
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3. Find the length of the hypotenuse given that
the legs of a right triangle are 6 ft and 12 ft.

1. 180 ft.
2. 324 ft.
3. 13.42 ft.
4. 18 ft.

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4. Find the length of the missing leg.

• 42 b2 102
• 16 b2 100
• -16 -16
• b2 84

The leg is about 9.2 cm long.
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5. Find the length of the missing leg.

• a2 122 132
• a2 144 169
• -144 -144
• a2 25

The leg is 5 inches long.
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6. Find the length of the missing side of a
right triangle if one leg is 4 ft and the
hypotenuse is 8 ft.

1. 24 ft.
2. 4 ft.
3. 6.9 ft.
4. 8.9 ft.

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Application of Pythagorean Theorem

• The screen aspect ratio, or the ratio of the
  width to the length of a HDTV is 169. The size
  of a television is

• given by the diagonal distance across the
  screen. If an HDTV is 41 inches wide, what is
  its diagonal screen size?
• What are the dimensions of a 65 inch HDTV?

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Application of Pythagorean Theorem

• A baseball diamond is a square with 90-foot
  sides. What is the approximate distance the
  catcher must throw from home to second base?

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The Converse of the Pythagorean Theorem
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The Converse of the Pythagorean Theorem

• A common application of the converse of the
• Pythagorean Theorem is used by carpenters to
• make sure a corner that they are constructing
  forms
• a right angle. Here are the steps
• Starting at the corner, measure 3 units along
• one direction and make a mark.
• 2. Measure 4 units along the other direction
  and make a mark.
• 3. Measure the distance between the marks.
• 4. If the length is equal to 5 units, then the
  corner forms a right angle (90)
• If the length is less than 5 units, then
  the corner is less than 90
• If the length is greater than 5 units, the
  corner is greater than 90
• Why? Since 32 42 52, then the triangle is a
  right triangle by the converse of the Pythagorean
  Theorem.

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5. The measures of three sides of a triangle are
given below. Determine whether the triangle is a
right triangle. , 3, and 8

• Which side is the longest?
• The square root of 73 is about 8.5, therefore it
  must be the hypotenuse.
• Plug your information into the Pythagorean
  Theorem. It doesnt matter which number is a or
  b.

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Sides , 3, and 832 82 ( ) 2

• 9 64 73
• 73 73
• Since this is true, the triangle is a right
  triangle!! If it was not true, it would not be a
  right triangle.

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Three right triangles surround a shaded triangle
together they form a rectangle measuring 12 units
by 14 units. The figure below shows some of the
dimensions but is not drawn to scale.
Is the shaded triangle a right triangle? Provide
proof for your answer.
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No, the shaded triangle is not a right triangle.
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The Distance Formula

• The distance formula is often memorized in the
  square root form with no connection to previous
  learning.
• Many students do not make the connection that the
  distance formula
• is simply the Pythagorean Theorem algebraically
  manipulated
• by solving for d, which is the
• hypotenuse of a right triangle..

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Deriving the Distance Formula
The Distance from Point A to Point B would be
equal to the length of the hypotenuse of triangle
ABC.
C
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Deriving the Distance Formula
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Find the Distance Between

1. Points A and B
2. Points B and C
3. Points A and C

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Roland went on a hike to visit a cave in the
mountains. To begin his hike he faced west and
hiked for 3 miles. Then he turned to the south
and traveled for 2 miles. After a water break
Roland again continued west for 4 miles. Turning
North he continued for 3 miles. Next Roland
turned left for 2 miles, and then he took a right
and continued on his hike for a final 6 miles
until he discovered the location of the cave.
As the crow flys, how far is the cave from
where Roland started his hike?
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West 3 miles South 2 miles West 4 miles North 3
miles Left 2 miles Right 6 miles
9
7
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Pythagorean Theorem in the 3rd Dimension
What is the longest curtain rod you can fit in
this box? Note Fishing poles only come in
increments of tenth of a foot.
?
?
2
?
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Pythagorean Theorem in the 3rd Dimension
What is the longest curtain rod you can fit in
this box? Note Fishing poles only come in
increments of tenth of a foot.
?
2
The longest rod is 5.3 feet
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Pythagorean Theorem in the 3rd Dimension
Derive a formula that will always work to find
the diagonal of any rectangular prism.