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.