Pythagorean Theorem and Special Right Triangles

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Pythagorean Theorem and Special Right Triangles
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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 other two sides.
• Given Triangle ABC with m(C) 90
• Prove a2 b2 c2
• Constructions
• Square on top of the hypotenuse
• Outside square where original triangle is a
  corner

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Prove a2 b2 c2

• Goal 1 Prove CHFD is a rectangle
• Need Parallelogram and 1 right angle
• Parallelogram Constructed sides to be parallel
• C is a right angle
• Goal 2 Show 4 Outer Triangles Congruent
• Each has a right angle
• Each has hypotenuse the same length
• Get a third angle congruent
• Goal 3 Set Areas Equal and Get Result
• Area of Big Square 4(Area of Triangle) (Area
  of Inner Square)

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Examples

• The legs of a right triangle are 3ft and 4ft in
  length. Find the length of the hypotenuse of the
  triangle.
• An isosceles-right triangle has legs of length 8.
  Find the length of its hypotenuse.
• The legs of a certain right triangle are
  congruent and the hypotenuse is 2v2. find the
  length of each leg of the triangle.
• The base of a 32-ft ladder is 7ft from the wall.
  The top of the ladder just reaches the bottom of
  the window. Find how far the bottom of the
  window is above the ground.
• Work on 5.3 1, 3, 5, 13, 19, 21, 23, 25, 29 and
  Page 1 of Worksheet

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Converse of Pythagorean Theorem

• Theorem If a triangle has sides of length a, b
  and c, and a2 b2 c2, then the triangle is a
  right triangle.
• Given Triangle with a2 b2 c2.
• Prove Triangle is right.
• Construct A right triangle with legs of the same
  lengths
• Goals
• Prove the triangles are congruent
• CPCTC gives right angle

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Fill In The Proof

1. ???
2. ???
3. a2 b2 d2
4. ???
5. ???
6. ???
7. ???
8. ???
9. ???
10. ???

1. Given
2. By Construction
3. ???
4. Substitution
5. Simplify
6. SSS
7. cpctc
8. Definition of Congruence
9. If an angle measures 90º, then it is a right
   angle
10. ???

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Examples

• A triangle has sides that measure 20, 21 and 29
  cm. Determine whether the triangle is a right
  triangle.
• Is a triangle with sides of 3, 7, and 11 meters a
  right triangle?
• Work on 5.3 7, 9, 11, 15, 17

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30-60-90 Triangle

• Theorem (Phrased differently) In a 30-60-90
  triangle, the hypotenuse is twice the length of
  the shorter leg. The longer leg is v3 times the
  shorter leg.
• Example A child, lying on level ground 43 ft
  from the base of a flagpole, sights its top at an
  angle of 30. Find the height of the flagpole.
• Example An equilateral triangle has sides 8 in
  long. Find its height.

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Proof of 30-60-90 Theorem

• Given
• Right triangle ABC with m(A) 30, m(B) 60.
  m(BC) a, m(AC) b, and m(AB) c.
• Prove
• c 2a and b a v3
• Construction
• Extend BC to point D so that m(BC) m(CD)
• Make triangle ADC

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Proof of 30-60-90 Theorem

• Prove the two triangles are congruent
• CPCTC shows the large triangle is equilangular
• Equiangular ? Equilateral ? c 2a
• Use Pythagorean Theorem
• Get b a v3

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Isosceles Right Triangle Theorem

• Theorem In an isosceles right triangle, the
  length of the hypotenuse is v2 times the length
  of one leg.
• Given Isosceles right triangle with legs of
  measure a and hypotenuse of measure c
• Prove c a v2
• Proof By Pythagorean Theorem

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Examples

• A square has sides 2v2 ft long. Find the length
  of its diagonal.
• Two boats are the same distance from a
  lighthouse, one due north and one due east. The
  boats are 17 nautical miles from each other. How
  far is each boat from the lighthouse?
• Work on Pages 2 and 3 of Worksheet

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Homework

• 5.3 1-25 odd, 29, 31
• 5.4 1-33 odd
• Worksheets
• Skipping 5.5 Chapter 5 notes due 3/31