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In the 6th century BC, a Greek philosopher named Pythagoras lived
in the village of Samos. He started a school where philosophy and
religion were studied, in addition to astronomy, mathematics and
music. The students from his school were called Pythagoreans. Central
to Pythagoras' teaching was the idea that all physical relationships
could be expressed by mathematical relations. One of the most famous
discoveries of the Pythagoreans was a proof for a distance relationship
that had been developed many years before by the Egyptians.
The Nile River flows through Egypt. This huge river is a source
of life in an otherwise barren, desert land. During the rainy season,
the Nile floods regularly. After each flood the surveyors would
have to reset the boundaries of the farmers' fields. Land was sectioned
into squares, so it was critical that the surveyors knew how to
mark a right angle (because squares have four right angles).
The clever Egyptians took a rope and tied twelve evenly-spaced
knots in it. They then made a triangle with the rope. One side had
three spaces between the knots, another had four spaces between
the knots, and the longest had five. This triangle was very special.
The angle opposite from the longest side was always a right angle.
Using this rope, the surveyors were able to show that the boundaries
they marked were indeed in the shape of a square.
Many years later, the Pythagoreans named a triangle that contained
a right angle, a "right triangle". They also named some of the parts
of a right triangle. They called the longest side, opposite the
right angle, the hypotenuse. The sides next to (or adjacent to)
the right angle were called the legs.
The Pythagoreans discovered that the legs and hypotenuse of a right
triangle did not always have to have lengths of 3, 4 and 5. But
the numbers did have to work in a special formula. The special formula
is called the Pythagorean Theorem. The Pythagorean Theorem goes
like this:
If you take the length of a leg of a triangle (say "a"
in the graphic above) and multiply it by itself (or "square" it);
a X a or a2
then, do the same with the length of the other leg ("b" in the
graphic above);
b X b or b2
and add the results together;
a2 + b2 =
your final result will be equal to the length of the hypotenuse
("c" in the graphic above) multiplied by itself (or "squared").
a2 + b2 = c2
The Pythagoreans also discovered that if they knew the lengths
of the two legs of a right triangle, they could use the Pythagorean
Theorem to find the length of the hypotenuse.
Say that one leg of a right triangle has a length of 6 units and
another has a length of 8 units. What is the length of the hypotenuse?
We know from the Pythagorean Theorem that
a2 + b2 = c2
In our example, a = 6 and b = 8. So,
a2 + b2 = c2
62 + 82 = c2
36 + 64 = c2
100 = c2
Since the square root of 100 is 10 (that is, 102 equals
100) the length of the hypotenuse must equal 10.
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