a2 + b2 = c2, Lecture notes of Piano

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THE PYTHAGOREAN

THEOREM

CFE 3285V

OPEN CAPTIONED

ALLIED VIDEO CORPORTATION

Grade Levels: 9-

14 minutes

a

+ b

= c

DESCRIPTION

Reviews the definition of a right triangle before a brief

history of the development of the Pythagorean theorem.

Shows how to use the theorem to solve for the

hypotenuse ( c ) and each of the two sides ( a and b ).

Briefly presents applications of the theorem.

INSTRUCTIONAL GOALS

• To review the right triangle and its parts.
• To re-create the development of the Pythagorean

theorem.

• To rewrite the Pythagorean theorem to solve for a

and b.

• To show how to use the Pythagorean theorem.
• To present practical applications of the

Pythagorean theorem.

BEFORE SHOWING

1. Read the CAPTION SCRIPT to determine

unfamiliar vocabulary and language concepts.

2. Using the calculator, practice finding the square

and square roots of numbers.

3. Make a time line from 585 BC to the present.

Point out the period of time in which Pythagoras lived.

DURING SHOWING

1. View the video more than once, with one showing

uninterrupted.

2. Pause at the scene showing the bust of

Pythagoras. His discovery has made him famous for

2600 years. Determine how that number was derived.

3. Pause at the scene showing the substitution of

numerical values for a and b.

a. Substitute other values for a and b and

calculate c.

b. Explain that the answer for c is usually a

radical expression, not an integer.

9. List some practical applications of the converse of

the Pythagorean theorem.

10. Report on the development of the square root.

Determine what methods were used to calculate square

roots in ancient times.

11. Design a work sheet with numerical values for a ,

b , and c. Determine which are values for a right

triangle.

WEBSITES

Explore the Internet to discover sites related to this

topic. Check the CFV website for related information

(http://www.cfv.org).

CAPTION SCRIPT

Following are the captions as they appear on the video.

Teachers are encouraged to read the script prior to

viewing the video for pertinent vocabulary, to discover

language patterns within the captions, or to determine

content for introduction or review. Enlarged copies

may be given to students as a language exercise.

[piano playing "Chopsticks"] (male narrator) Welcome to the assistant professor's educational video series. Today, we are studying: The first lesson defines the Pythagorean Theorem. The second lesson will solve some example problems, and explain how to use the theorem. Before we begin, let's review the definitionof a right triangle.

A right triangle has one 90 degree, or right, angle. Usually, the sides of a right triangle are named: Side "c" is called the hypotenuse. The hypotenuse is always the side opposite the right angle. Sides "a" and "b" are simply called sides. The sides meet to form the 90 degree angle. This object is a natural, perfect right triangle.

Its sides always measure 3 units, 4 units, and its hypotenuse, 5 units. The ancient Greeks discovered this natural right triangle, the builder's triangle. They used it to construct houses, buildings, and temples. Using the builder's triangle, they could insure that their corners, joints, and vertices were perfect 90 degree angles. In about 600 B.C.,on the Greek island of Samos, there lived a man named Pythagoras. Pythagoras was a mathematician and a philosopher. Pythagoras was fascinated by geometry, and he was interested in triangles. He noticed something interesting about the shape and measurements of the builder's triangle. He made a square out ofthe length of each side

of both sides of the equality. Taking the square root of a value is the reverse process of squaring a value. The square root of "c" squaredis "c."

The square root of 25 is 5. The value of "c," or the length of the hypotenuse, is 5 units. We know the value for each side of this triangle. That was pretty easy. The Pythagorean Theorem works. Pythagoras' theorem is also versatile. We can manipulate itto create a formula

that solves for the length of the hypotenuse. For any right triangle whose sides are labeled "a" and "b," and whose hypotenuse is "c," the length of "c" is equal to the square root of "a" squared plus "b" squared. This expression states that we are solving for the length of "c," the hypotenuse. We substitute the valuesin the formula,

and solve for the value of "c." We know how to find the length of the hypotenuse,

but how can we find side "b?" For this right triangle, suppose we know the length of side "a," and the length of the hypotenuse, or side "c." How do we solve for the lengthof side "b?" Again, we can manipulate Pythagoras' famous theorem: To isolate the value of "b," we subtract "a" squared from both sides of the equality. This leaves us with the equality: To find the value of side "b," we take the square root of each side of the equality. The square root of "b" squared is "b." The value of "b" equals the square root of the expression: Let's use this expression to find side "b" of this triangle. Side "c" is 4 units. Side "a" is 3 units. Using the formula, substitute the values for "c" and "a." 4 squared equals 16, and 3 squared equals 9. 16 minus 9 equals 7. "b" equals thesquare root of 7. Unfortunately, Pythagoras did not have a calculator, so he had to do a lot of arithmetic.

We use the square root function on a calculator, and find that the square root of 7 equals 2.646. The length of side "b" is 2.646 units. Let's move on to another example. We know how to find two of the sides of a right triangle. How can we find the third side, "a?" We can begin with the original theorem, but we need to rearrange it to solve for the value of "a." To isolate the value of "a," subtract the value of "b" squared from both sides of the equality. Since we are solving for the value of "a," take the square root of both sides of the equality. The value of side "a" equals the square root of "c" squared minus "b" squared. In this example, let's suppose that side "c" is 7 units long and that side "b" is 5 units long. Using the formula, substitute the values in the equation, and then solve. 7 squared is 49, and 5 squared is 25. 49 minus 25 equals 24.

The length of side "a" equals the square root of 24. Using a calculator, the length of side "a" is about 4.899 units. Let's review what we learnedabout the Pythagorean Theorem. or any right triangle whose sides are named "a" and "b," and whose hypotenuse is named "c:" The length of the hypotenuse, or side "c," is: The length of side "a" is: The length of side "b" is: What's so important about Pythagoras? Something must be important since we remember him after 3,000 years. Pythagoras is the first known person to develop the idea that the physical world can be modeled and represented through pure mathematics. Pythagoras showed us that mathematics is a special kind of language we can use to develop ideas, predict outcomes, and solve problems. Problems can be solved mathematically on paper instead of relying on tedious experimentation, or trial and error.