Transcript Lesson 7.2

Bellringer
O Simplify each expression
1.
5 ∙ 10
2.
243
3.
6∙ 8
7-2 The Pythagorean
Theorem and Its
Converse
Pythagorean Theorem (Thm. 7-4): 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.
Pythagorean Theorem
2
2
𝑎 +𝑏 =𝑐
Leg
Leg
2
Hypotenuse
Pythagorean Triple: set of nonzero
whole numbers, a, b, and c that satisfy
the Pythagorean Theorem
Common Triples:
3, 4, 5
5, 12, 13
7, 24, 25
8, 15, 17
12, 25, 37
20, 21, 29
2
2
2
𝑎 +𝑏 =𝑐
82 + 𝑏 2 = 122
64 + 𝑏 2 = 144
𝑏 2 = 80
𝑏 2 = 80
𝑏=4 5
Bellringer
Find the value of x
12
x
x
12
15
8
𝑎2 + 𝑏2 = 𝑐 2
2502 + 3502 = 𝑐 2
62,500 + 122,500 = 𝑐 2
185,000 = 𝑐 2
185,000 = 𝑐 2
185,000 = 𝑐
𝑐 = 430.11626
𝑐 ≈ 430𝑚
Finding the Area using the Pythagorean Theorem
Use the
Pythagorean
Thm. to find
the height
first!
12 m
12 m
102 + ℎ2 = 122
100 + ℎ2 = 144
ℎ2 = 44
ℎ2 = 44
ℎ = 2 11
20 m
𝐴 = 𝑏ℎ
1
𝐴 = (20)(2 11)
2
𝐴 = 20 11
Converse of the Pythagorean Theorem (Thm. 7-5): If the
square of the length of one side of a triangle is equal to
the sum of the squares of the lengths of the other two
sides, then the triangle is a right triangle.
𝑎2 + 𝑏2 = 𝑐 2
42 + 62 = 72
16 + 36 = 49
52 ≠ 49
Since the sum of the square of
the legs is not equal the
square of the hypotenuse, this
cannot be a right triangle.
𝑎2 + 𝑏2 = 𝑐 2
132 + 842 = 852
169 + 7,056 = 7,225
7,225 = 7,225
Since the sum of the square of
the legs is equal to the square
of the hypotenuse, this is a right
triangle.
Theorem 7-6:
If c  a  b , then the
triangle is obtuse.
2
2
2
Theorem 7-7: If c 2  a 2  b 2 , then the
triangle is acute.
The lengths of the sides of a triangle are given. Classify
each triangle as acute, obtuse, or right.
𝑐 2 = 𝑎2 + 𝑏2
142 = 62 + 112
196 = 36 + 121
196 = 157
196 > 157
Since the square of the hypotenuse
is greater than the sum of the square
of the legs, the triangle is an obtuse
triangle.
The lengths of the sides of a triangle are given. Classify
each triangle as acute, obtuse, or right.
𝑐 2 = 𝑎2 + 𝑏2
152 = 122 + 132
225 = 144 + 169
225 = 313
225 < 313
Since the square of the hypotenuse
is less than the sum of the square
of the legs, the triangle is an acute
triangle.
Assignment
Pg. 361 #’s 2 – 34 even