9.2 the pythagorean theorem geometry mrs. gibson spring 2011
TRANSCRIPT
9.2 The Pythagorean 9.2 The Pythagorean TheoremTheorem
GeometryMrs. GibsonSpring 2011
Objectives/AssignmentObjectives/Assignment
GLE 12: Apply the Pythagorean Theorem in both abstract and real-life settings
Objectives:Prove the Pythagorean TheoremUse the Pythagorean Theorem to solve
real-life problems such as determining how far a ladder will reach.
Assignment: pp. 538-539 #1-31 odd
Proving the Pythagorean TheoremProving 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.
History LessonHistory 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.
Theorem 9.4: Pythagorean Theorem 9.4: Pythagorean TheoremTheorem 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
hyp2 = leg2 + leg2
Using the Pythagorean TheoremUsing 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 Ex. 1: Finding the length of the hypotenuse.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: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
Using the Pythagorean TheoremUsing the Pythagorean Theorem
List of Common Pythagorean Triples:◦Multiples of 3,4,5 = 6,8,10 9,12,15
21,28,35◦Multiples of 5,12,13 = 10,24,26
25,60,65◦Multiples of 8,15,17 = 16,30,34
24,45,51◦Multiples of 7,24,25 = 14,48,50
21,72,75◦Multiples of 9,40,41 = 18,80,82
27,120,123
Ex. 2: Finding the Ex. 2: Finding the Length of a LegLength of a LegFind the length of the leg of the right triangle.
x
14
7
Solution: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
Ex. 3: Finding the Ex. 3: Finding the area of a trianglearea of a triangleFind 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: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 of a Triangle
Area = ½ bh = ½ (10)(√24)
≈ 24.5 m2
The area of the triangle is about 24.5 m2
h
10 m
7 m7 m
Ex. 4: Indirect Ex. 4: Indirect MeasurementMeasurementSupport Beam: The skyscrapers
shown on page 535 are connected by a skywalk with support beams. You can use the Pythagorean Theorem to find the approximate length of each support beam.
Each support beam forms the hypotenuse of a right triangle. The right triangles are congruent, so the support beams are the same length. Use the Pythagorean Theorem to show the length of each support beam (x).
Solution:Solution:
(hypotenuse)2 = (leg)2 + (leg)2
x2 = (23.26)2 + (47.57)2
x2 = √ (23.26)2 + (47.57)2
x ≈ 53
Pythagorean Theorem
Substitute values.
Multiply and find the positive square root.
Use a calculator to approximate.