Lovin’ Those Right Triangles!
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
In most of the ancient civilizations that
concerned themselves with mathematics,
individuals discovered a relationship with
sides of a right triangle.
Pythagoras was credited with the theorem,
even though it was probably discovered long
before him.

In a right triangle with legs a and b, and
hypotenuse c,
c
a
b
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Here are some visual proofs of the
Pythagorean theorem: Let’s check ‘em out!
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If you know the two legs, you can plug in the
values for a and b.
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Square both of them
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Add them together
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Take the square root of the sum.
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A rectangle has sides of 6 and 8. How long is
the diagonal?
◦ A rectangle is just two right triangles put together
right?
6
8
◦ So for this problem, a = 6, b = 8, and we’ll let c = x.
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So if a = 6, b = 8, and c = x, when we
2
2
2
substitute into a  b  c , we get
6 8  x
2
2
2
36  64  x
100  x 2
10  x
2
*Now, to solve for x, we’ll
square the 6 and 8.
*We can add the 36 and 64.
*Finally, to undo squaring, we
take the square root of each
side
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A right triangle has legs of length 3 and 4.
What is the length of the hypotenuse?
Remember to draw your figure!
x
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First we draw the figure:
Now we find our a, b, and c.
4
3
◦ 4 and 3 are legs, so they can be either a or b.
◦ We’ll say a = 4, b = 3, and c = x
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Substitute into the theorem and get:
◦
4 3  x
2
2
2
 Try solving it on your own!
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You should get x = 5
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This is slightly harder
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First, plug in c and either a or b.
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Then square them.
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You’ll then have to subtract to get the
missing leg by itself
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Finally, take the square root.
Find the length of the missing side:
◦ For this problem, a = x, b = 25, & c = 27
◦ Substitute them into a 2  b 2  c 2 and get
25 cm

a
x 2  252  27 2
*square the 25 and 27
x 2  625  729
*To solve for x, subtract the 625
x 2  104
*Now take the square root.
Since it isn’t perfect, round to
the tenths place
x  10.2cm

If a right triangle has a hypotenuse of 14 in
and one of the legs is 9 in, find the third leg.
◦ Try this one on your own.
◦ Start by drawing the figure.
◦ Set a = x, b = 9, and c = 14.
◦ Solve it like the last problem.
◦ You should get x = 10.7 in.