The Converse of the Pythagorean Theorem

1
Lesson 8-3

• The Converse of the Pythagorean Theorem

2
Objectives

• State and apply the converse of the Pythagorean
  Theorem and related theorems about obtuse and
  acute triangles.

3
BIBLICAL INTEGRATION

• Man's ability to understand and work with
  numbers, as well as with other subjects,is a gift
  from God.
• (Job 32 8,9 I Cor. 214-16)

4
The Pythagorean Theorem says
A

• If ?ABC is a right triangle,
• then a² b² c².

c
b
B
C
a

• The converse is also true
• If a² b² c², then ?ABC is a right
  triangle.

5
Theorem 8-3

• If the square of one side of a triangle is equal
  to the sum of the squares of the other two sides,
  then the triangle is a right triangle.

A
c
b
B
C
a
If c a b, then ?C is a right angle and ?ABC
is a right triangle.
6
Example 1

• If a triangle is formed with sides having the
  lengths given, is it a right triangle?

a. 16, 30, 34
b. 3, 4, 6
c. 1, 7, 7
16² 30² ? 34²
3² 4² ? 6²
1² 7² ? 7²
1 49 ? 49
256 900 ? 1156
9 16 ? 36
1156 1156
25 lt 36
50 gt 49
Yes. Its a rt. ?
No. Its not a rt. ?
No. Its not a rt. ?
7
Individual Practice

• 1. 4, 7, 9 2. 20, 21, 29

4² 7² ? 9² 16 49 ? 81 65 lt 81
20² 21² ? 29² 400 441 ? 841 841
841
No. rt. ?
Yes. rt. ?
3. v2, 2, v5 4. 0.8, 1.5, 1.7
(v2 )² 2² ? (v5)² 2 4 ? 5
6 gt 5
0.8² 1.5² ? 1.7² 0.64 2.25 ? 2.89
2.89 2.89
No. rt. ?
Yes. rt. ?
8
Pythagorean Triples

• A triangle with sides of 3, 4, and 5 is a right
  triangle because 3² 4² 5².
• Any triangle with sides 3n, 4n, and 5n, n gt 0,
  is also a right triangle because (3n)²
  (4n)² (5n)²
• Multiples of any three lengths that form a right
  triangle will also form right triangles.
• These groups of three lengths are called
  Pythagorean Triples.
• 3, 4, 5 5, 12, 13 8, 15, 17 7, 24, 25
• 6, 8, 10 10, 24, 26 16, 30, 34
  14, 48, 50
• 9,12,15 15, 36, 39 24, 45, 51
  21, 72, 75

9
Find the value of x.
48
x
6.

• 5.

14
x
16
34
x 50
x 30
7.
8.
65
50
40
25
x
x 60
x
x 30
10
Theorem 8-4

• If the square of the longest side of a triangle
  is greater than the sum of the squares of the
  other two sides, then the triangle is an obtuse
  triangle.

A
c
b
B
a
C
If c² gt a² b², then ?C is obtuse and ?ABC is
an obtuse triangle.
11
Theorem 8-5

• If the square of the longest side of a triangle
  is less than the sum of the squares of the other
  two sides, then the triangle is an acute
  triangle.

A
c
b
B
a
C
If c² lt a² b², then ?C is acute and ?ABC is an
acute triangle.
12
Example 2

• a. 8, 12, 13 b. 4, 4, 7

7² ? 4² 4²
13² ? 8² 12²
169 ? 64 144
49 ? 16 16
169 lt 208
49 gt 32
Lengths acute triangle
Lengths obtuse triangle
13
Individual Practice

• 9. 8, 9, 12 10. v5, v5, v10

(v10)² ? (v5) ² (v5)² 10 ? 5 5
10 10
12² ? 8² 9² 144 ? 64 81 144 lt 145
Lengths acute
Lengths rt. angle
14

• 11. 8, 13, 20 12. 5, 7, v74

(v74)² ? 5² 7² 74 ? 25 49 74
74
20² ? 8² 13² 400 ? 64 169 400 gt 233
Lengths obtuse triangle
Lengths rt. triangle
13. 2, 2v3, 4 14. 8, 11, 15
4² ? 2² (2 v3)² 16 ? 4 (4 3) 16 ? 4 12 16
lt 16
15² ? 8² 11² 225 ? 64 121 225 gt 185
Lengths obtuse triangle
Lengths rt. triangle
15
Individual Practice (contd)

• 15. 4, 5, 6 16. 5, 5, 5 v3

6² ? 4² 5² 36 ? 16 25 36 lt 41
(5 v3) ² ? 5² 5² (25 3) ? 25 25 75
gt 50
Lengths obtuse triangle
Lengths acute triangle
16
If each diagram were drawn to scale, which
angle(s) would be right angles?
B
13
F
G
30
12
16
5
17
12
C
5
I
A
D
17
17
E
17.
H
18.
?ABC 16, 30, 34
?FIG 5, 12, 13 ?FIE 5, 12, 13 ?GIH 5, 12,
13 ?HIE 5, 12, 13
17
19.

• ?JML 8, 15, 17

20.
?NQO 8, 15, 17 ?QOP 15, 20, 25
18
Homework

• (p. 297 1-14)