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12.3 The Pythagorean
TheoremCORDMrs. Spitz
Spring 2007
Objectives/Assignment
•Use Pythagorean Theorem
•Assignment: pp. 484-485 #4-39 all
•Assignment due today: 12.2 p. 479-480 #5-51
History Lesson•Around the 6th century BC, the Greek mathematician Pythagorus founded a school for the study of philosophy, mathematics and science. Many people believe that an early proof of the Pythagorean Theorem came from this school.
•Today, the Pythagorean Theorem is one of the most famous theorems in geometry. Over 100 different proofs now exist.
Proving the Pythagorean
Theorem•In this lesson, you will study one of the most famous theorems in mathematics—the Pythagorean Theorem. The relationship it describes has been known for thousands of years.
Theorem 9.4: Pythagorean
Theorem•In a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the legs.
ca
b
c2 = a2 + b2
Using the Pythagorean
Theorem•A Pythagorean triple is a set of three positive integers a, b, and c that satisfy the equation c2 = a2 + b2 For example, the integers 3, 4 and 5 form a Pythagorean Triple because 52 = 32 + 42.
Ex. 1: Finding the length of the hypotenuse.•Find the length of the hypotenuse of the right triangle. Tell whether the sides lengths form a Pythagorean Triple.
x
12
5
Solution:(hypotenuse)2 = (leg)2 + (leg)2
x2 = 52 + 122
x2 = 25 + 144 x2 = 169
x = 13
Because the side lengths 5, 12 and 13 are integers, they form a Pythagorean Triple. Many right triangles have side lengths that do not form a Pythagorean Triple as shown next slide.
Pythagorean TheoremSubstitute values.MultiplyAddFind the positive square
root.
Note: There are no negative square roots until you get to Algebra II and introduced to “imaginary numbers.”
x
12
5
Ex. 2: Finding the Length of a Leg
•Find the length of the leg of the right triangle.
x
14
7
Solution:(hypotenuse)2 = (leg)2 + (leg)2
142 = 72 + x2
196 = 49 + x2
147 = x2
√147 = x
√49 ∙ √3 = x
7√3 = x
Pythagorean Theorem
Substitute values.
Multiply
Subtract 49 from each side
Find the positive square root.
Use Product property
Simplify the radical.
x
14
7
In example 2, the side length was written as a radical in the simplest form. In real-life problems, it is often more convenient to use a calculator to write a decimal approximation of the side length. For instance, in Example 2, x = 7 ∙√3 ≈ 12.1
Note:•Determine if the following lengths can represent the sides of a right triangle.
c2 = a2 + b2 Right ∆
c2 > a2 + b2
c2 < a2 + b2 Acute ∆
Obtuse ∆
#32. 12, 11, 15•The measures of the sides of a triangle are given. Determine whether each triangle is a right triangle.
c2 = a2 + b2
152 = 112 + 122
225 = 121 +144?225 ≠ 265 Not a right Triangle
#14 a = √7 b =√9 c=?
c2 = a2 + b2
c2 = √72 + √92
c2 =7 + 9c2 = 16c = 4
Note:
7
49
7772
Area of a rectangle
•The area of a rectangle is 40 square meters. Find the length of a diagonal of the rectangle if its length is 2 meters less than twice its width.
40 = bhb = 2h
+ 2
?
Ex. 3: Finding the area of a triangle
•Find the area of the triangle to the nearest tenth of a meter.
•You are given that the base of the triangle is 10 meters, but you do not know the height.
h
10 m
7 m7 m
Because the triangle is isosceles, it can be divided into two congruent triangles with the given dimensions. Use the Pythagorean Theorem to find the value of h.
Solution:Steps:
(hypotenuse)2 = (leg)2 + (leg)2
72 = 52 + h2
49 = 25 + h2
24 = h2
√24 = h
Reason:
Pythagorean Theorem
Substitute values.
Multiply
Subtract 25 both sides
Find the positive square root.
h
10 m
7 m7 m
Now find the area of the original triangle.
Area of a TriangleArea = ½ bh = ½ (10)(√24)
≈ 24.5 m2
The area of the triangle is about 24.5 m2
h
10 m
7 m7 m