9.2 The Pythagorean Theorem

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9.2 The Pythagorean Theorem. Geometry Mrs. Spitz Spring 2005. Objectives/Assignment. Prove the Pythagorean Theorem Use the Pythagorean Theorem to solve real-life problems such as determining how far a ladder will reach. Assignment: pp. 538-539 #1-31 all

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### 9.2 The Pythagorean Theorem

Geometry

Mrs. Spitz

Spring 2005

Objectives/Assignment
• Prove the Pythagorean Theorem
• Use the Pythagorean Theorem to solve real-life problems such as determining how far a ladder will reach.
• Assignment: pp. 538-539 #1-31 all
• Assignment due today: 9.1 pp. 531-532 #1-34 all
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.
In a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the legs.Theorem 9.4: Pythagorean Theorem

c2 = a2 + b2

Proving the Pythagorean Theorem
• There are many different proofs of the Pythagorean Theorem. One is shown below. Other proofs are found in Exercises 37 and 38 on page 540 and in the Math & History feature on page 557.

Given: In ∆ABC, BCA is a right angle.

Prove: c2 = a2 + b2

Proof:

Plan for proof: Draw altitude CD to the hypotenuse just like in 9.1. Then apply Geometric Mean Theorem 9.3 which states that when the altitude is drawn to the hypotenuse of a right triangle, each leg of the right triangle is the geometric mean of the hypotenuse and the segment of the hypotenuse that is adjacent to that leg.

Statements:

1. Draw a perpendicular from C to AB.

2. and

Reasons:

Proof
• Perpendicular Postulate
• Geometric Mean Thm.
• Cross Product Property

c

a

c

b

=

=

a

e

b

f

3. ce = a2 and cf = b2

4. ce + cf = a2 + b2

5. c(e + f) = a2 + b2

5. Distributive Property

6. e + f = c

7. c2 = a2 + b2

7. Substitution Property of =

Using the Pythagorean Theorem
• A Pythagorean triple is a set of three positive integers a, b, and c that satisfy the equation c2 = a2 + b2For example, the integers 3, 4 and 5 form a Pythagorean Triple because 52 = 32 + 42.
(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 Theorem

Substitute values.

Multiply

Find the positive square root.

Note: There are no negative square roots until you get to Algebra II and introduced to “imaginary numbers.”

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

Solution:
• 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

You are given that the base of the triangle is 10 meters, but you do not know the height.

Ex. 3: Finding the area of a triangle

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.

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.

Solution:

Now find the area of the original triangle.

Area of a Triangle

Area = ½ bh

= ½ (10)(√24)

≈ 24.5 m2

The area of the triangle is about 24.5 m2

Support 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. Ex. 4: Indirect Measurement
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).
(hypotenuse)2 = (leg)2 + (leg)2

x2 = (23.26)2 + (47.57)2

x2 = √ (23.26)2 + (47.57)2

x ≈ 13

Pythagorean Theorem

Substitute values.

Multiply and find the positive square root.

Use a calculator to approximate.

Solution:
Reminder:
• Friday is the last day to turn in assignments for credit for the quarter.
• Quiz after 9.3 on Monday 3/10/05.
• Quiz after 9.5 probably Thursday (5th) and Friday (2nd and 6th).
• Test on Chapter 9 will be before you go on Spring Break. Do yourself a favor and take it before you go.