Objective: To state and apply the Pythagorean Theorem and its converse

GA QCC: #24

Lesson: Pythagorean Theorem

Name: ___________________________

HISTORICAL SUMMARY:

Pythagoras of Somas (located in Greece) lived from about 569 to 475 BC. He is often described as the first pure mathematician. We particularly remember him for a famous geometry theorem that bears his name. Although the mathematical relationship was known to the Babylonians about 1000 years earlier, Pythagoras is credited for proving it. Pythagoras should also be credited with reducing mathematics to the abstract. For example the number 2 may refer to the number of ships, houses, writing implements, and coins of any denomination.The manipulations with the abstract quantities (numerical representations) is valid for all the material things for which it represents. Hence, in my opinion, Pythagoras helped pave the way for modern algebra.

He was a world traveler (his travels to Egypt helped his study of mathematics); political activist (which caused him to get into numerous political scrapes including being a prisoner on a foreign soil); philosopher (the world and cosmos could be reduced to numbers, the world dynamics consisted of opposites, successive reincarnation through different species, and the brain is the locus of the soul); musician(an accomplished lyre player who contributed mathematics to music theory and understood the soothing power of music to the sick); and founder of two mathematics schools: one in Samos, "The Semicircle of Pythagoras" (which became political and exists to this day), and the other school (devoid of politics) was a Mathematikoi Society in Croton (now Italy) that is credited for outstanding contributions in mathematics.

There are 6 major contributions/theorems attributed to Pythagoras (or the Pythagoreans):

1. The sum of angles of a triangle equals 2 right angles, and the generalization of that theorem that addresses the sum of interior and exterior angles of a polygon.

2. Proving the theorem about right triangles in which the square of the hypotenuse equals the squares of the other two sides.

3. Solving algebraic equations with geometry such as

4. Discovering irrational numbers.

5. Discovering the 5 regular solids.

6. Even though Pythagoras thought the earth was the center of the universe (typical at that time), he recognized that (a) the orbit of the moon was inclined to the earth's equator, and (b) the planet Venus was both the morning star and the evening star in the sky.

For more detailed Pythagoras biography, CLICK HERE. (http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Pythagoras.html, retrieved August 11, 2000 from the World Wide Web)

PYTHAGOREAN THEOREM:

1. The Pythagorean Theorem tells us that, in a right triangle, the square of the hypotenuse equals the squares of the other 2 sides.

2. As a visual representation, study the figure below:

Since angle BAC is a right angle, then the smaller areas BFGA (blue) and AHIC (green) total 1.6 square inches - the value of area BCDE, 1.6 square inches. This relationship holds true as long as angle BAC is a right angle.

CLICK HERE if you wish to experiment on your own. Only change the lengths of the two legs; do NOT change angle A.

3. As a mathematical representation, the Pythagorean Theorem is usually expressed as

where c is the length of the hypotenuse of the right triangle, a and b are the 2 other sides which are called "legs" (of a right triangle).

4. Without reviewing the above material, state the Pythagorean Theorem in your own words:

5. Problems:

a. If b=3 and c=5, what is a? a = ____________

b. If a=4 and b= , what is c? c= __________

c. If c=2 and a=1, what is b? b= ______________

d. A 25 foot ladder is leaning against the wall. The base of the ladder is 10 feet from the wall. How far up the wall will the ladder reach? (Draw a picture and label it.)

Answer: ________________________

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