Question 72631
The Pythagorean theorem says that the sum of the squares of the two legs (short sides) of
a right triangle are equal to the square of the long side (the hypotenuse). In equation
form this is:
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{{{a^2 + b^2 = c^2}}}
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in which "a" and "b" represent the legs and "c" represents the hypotenuse.  For this problem
we know that:
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{{{a = 3}}} 
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and:
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{{{b = 4}}}
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Substitute these values into the equation for the Pythagorean theorem and you get:
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{{{3^2 + 4^2 = c^2}}}
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Now do the calculations. {{{3^2 = 3*3 = 9}}} and {{{4^2 = 4*4 = 16}}}. Substitute 
these values to get:
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{{{9 + 16 = c^2}}}
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The addition on the left side gives you:
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{{{25 = c^2}}}
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and you now take the square root of both sides to get:
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{{{c = 5}}}
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So the hypotenuse or long side of this right triangle is 5 units long.
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Just for information, any builder knows about the 3 - 4 - 5 triangle.  He or she uses it
to make sure that walls are at right angles. They do this by measuring down one wall 3 feet 
and marking that spot. They then measure down the perpendicular wall 4 feet and marking
that spot also.  Then they measure the slant distance between the two spots.  If that 
distance isn't 5 feet exactly, then they know the walls are not really perpendicular to each
other and they will have to adjust the walls until they get the distance between the 3-foot
and 4-foot marks to be exactly 5 feet.  Sometimes geometry really is helpful in the real
world.
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Remember this one.  Someday you're going to want to fence something in and you can use
this triangle to make sure the corners of your fence lines are a right angles to each other
(even if you can't remember what the Pythagorean theorem is).
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Hope this helps you to understand the Pythagorean theorem a little better.