5.7A The Pythagorean Theorem
Objectives:
G.SRT.8: Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied
problems.
For the Board: You will be able to use the Pythagorean Theorem and its converse and its inequalities
to solve problems.
Anticipatory Set:
A Pythagorean Triple is a set of three positive integers a, b, and c that satisfy the equation
c2 = a2 + b2.
3, 4, 5 is an example of a Pythagorean Triple.
32 + 42 = 9 + 16 = 25 = 52
0.5, 1.2, 1.3 is not an example of a Pythagorean Triple, even through 0.52 + 1.22 = 1.32, because these
numbers are not positive integers.
2, 3, 4 is not an example of a Pythagorean Triple, because 22 + 32 ≠ 42.
White Board Activity:
Practice: Which of the following are Pythagorean Triples?
a. 5, 12, 13
b. 4, 5, 9
2
2
5 + 12 = 25 + 144 = 169 = 132
42 + 52 = 16 + 25 = 41 which is not 92 (81)
82 + 152 = 64 + 225 = 289 = 172
c. 8, 15, 17
Yes
No
Yes
Instruction:
Open the book to page 362 and read example 3.
Example: Find the missing side length, then tell if the side lengths form a Pythagorean triple.
a.
b.
4
14
48
142 + 482 = 196 + 2304 = c2
c2 = 2500
c = 50
Yes
12
42 + b2 = 122
b2 = 128
No
16 + b2 = 144
b = 128  2  64  8 2
White Board Activity:
Practice: Find the missing side length, then tell if the side lengths form a Pythagorean triple.
a.
b.
x2 + 242 = 262
24
10
8
x2 = 576 = 676
x2 = 100
26
82 + 102 = x2
x = 10
64 + 100 = x2
yes
2
164 = x
x = 12.8
No
Pythagorean Inequalities Theorem
In ΔABC, c is the length of the longest side.
If c2 < a2 + b2, then ΔABC is an acute triangle.
If c2 = a2 + b2, then ΔABC is a right triangle.
If c2 > a2 + b2, then ΔABC is an obtuse triangle.
a
c
b
a
c
b
c
a
b
Recall: Triangle Inequality Theorem
The sum of the measures of the two smaller sides must be larger than the measure of the
third side.
Example: 4, 8, 9 form a triangle because 4 + 8 = 12 > 9
3, 4, 7 do not form a triangle because 3 + 4 = 7 which is not > 7
2, 5, 12 do not form a triangle because 2 + 5 = 7 which in not > 12
Open the book to page 363 and read example 4.
Example: Tell if the measures con be the side lengths of a triangle. If so classify the triangle as acute,
obtuse, or right.
A. 7, 8, 9
B. 8, 8, 15
7 + 8 = 15 > 9
8 + 8 = 16 > 15
Yes, triangle.
Yes, triangle.
92
72 + 82
152
8 2 + 82
81
49 + 64
225
64 + 64
81
113
225
128
81 < 113, the triangle is acute.
225 > 128,the triangle is obtuse.
White Board Activity:
Practice: Tell if the measures can be the side lengths of a triangle. If so classify the triangle as acute,
obtuse, or right.
a. 5, 7, 10
b. 5, 8, 17
5 + 7 = 12 > 10
5 + 8 = 13 < 17
Yes they form a triangle
No does not form a triangle
102
52 + 72
100
25 + 49 = 74
Since 100 > 74 the triangle is obtuse
Since practice prob. b does not form a triangle, change one of the sides so that it does form a triangle.
Then determine whether it is acute, obtuse, or right.
Example: Change the 8 to 15, then 5 + 15 = 20 which is > 17.
172
52 + 152
289
25 + 225 = 250
Since 289 > 250 the triangle is obtuse.
Assessment:
Question student pairs.
Independent Practice:
Text: pgs. 364 - 367 prob. 6 – 14, 19 – 27.
For a Grade:
Text: pgs. 364 – 367 prob. 8, 10, 20, 22.