Solving Problems with Geometric ModelsPythagorean Theorem

1
Solving Problems with Geometric
Models-Pythagorean Theorem

• Math- Geometry
• High School

2
Teacher Page

• Content Pythagorean Theorem
• Grade Level 8-12 (Algebra I and Geometry)
• Creator Don Sutcliffe
• Other Copy slides 15-24 as a handout for
  students to take notes.

3
Teacher Page

• Curriculum Objectives
• Geometry III.B Solve for the missing side
  lengths in a right triangle using the Pythagorean
  Theorem.
• Curriculum Objectives
• Missouri Show-Me Standard Math MA1, MA2, MA4,
  MA5
• Missouri Show-Me Standard Goal G3.1, G3.4,
  G3.5

4
In a right triangle

• Legs
• The short sides
• Opposite the acute angles
• Sides a and b
• Hypotenuse
• Longest side
• The side opposite the right angle
• Side c

A
b
c
B
C
a
5
Pythagorean Theorem

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

A
b
c
a2 b2 c2
B
C
a
6
Proof of the Pythagorean Theorem
a
b
Area of the center square c2
b
c
a
c
Area of the ?s 4(1/2ab) 2ab
c
a
c
b
Area of large square (ab)2 a22abb2
a
b
7
ProofContinued
a
b
b
c
a
c
c
a
c
b
b
a

-
Area of the center square
Area of large square
Area of the ?s
-
2ab
c2
a22abb2


a2 b2
c2
8
Find the length of the third side of the given
triangle
B

• a12, b9

12292c2 14481c2 225c2 15c
c
12
9
A
C
9
Find the length of the third side of the given
triangle

• b5, c13
• a12
• a3, c 2v6
• bv15

10
Converse of the Pythagorean Theorem

• If a, b, and c are lengths of the sides of a
  triangle, and a2b2c2, then the triangle is a
  right triangle.
• Used to test if a triangle is a right triangle.

11
Test to see if the triangle is a right triangle.

• 3, 4, 5
• 3242 ? 52
• 916 ? 25
• 25 25
• Right Triangle

• 8, 10, 12
• 82102 ? 122
• 64100 ? 144
• 164 ? 144
• Not A Right Triangle

12
Test to see if the triangle is a right triangle.

• 5, 13, 12

• 4, 5, 6

• Not A Right Triangle

• Right Triangle

13
If a tower is 100 feet tall and a guy wire is to
be placed 75 feet from the base of the tower on
all four sides, how much cable is needed for the
project?

• 1002752c2
• 10,0005625c2
• 15,625c2
• v15,625vc2
• 125c
• 125?4 sides 500 feet of cable

Guy Wire
Tower 100 feet
75 feet
14
End Show
15
In a right triangle

• Legs
• Hypotenuse

A
b
c
B
C
a
16
Pythagorean Theorem

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

A
b
c
B
C
a
17
Proof of the Pythagorean Theorem
a
b
Area of the center square
b
c
a
c
Area of the ?s 4(1/2ab)
c
a
c
b
Area of large square (ab)2
a
b
18
ProofContinued
a
b
b
c
a
c
c
a
c
b
b
a

-
Area of the center square
Area of large square
Area of the ?s
19
Find the length of the third side of the given
triangle
B

• a12, b9

c
12
9
A
C
20
Find the length of the third side of the given
triangle

• b5, c13
• a3, c 2v6

21
Converse of the Pythagorean Theorem

• If a, b, and c are lengths of the sides of a
  triangle, and a2b2c2, then the triangle is a
  right triangle.

22
Test to see if the triangle is a right triangle.

• 3, 4, 5

• 8, 10, 12

23
Test to see if the triangle is a right triangle.

• 5, 13, 12

• 4, 5, 6

24
If a tower is 100 feet tall and a guy wire is to
be placed 75 feet from the base of the tower on
all four sides, how much cable is needed for the
project?
Guy Wire
Tower 100 feet
75 feet