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4. Important Special Triangles
Two sides of a triangle have equal length if and only if the angles opposite them have equal measure. Such a triangle is called an isosceles triangle. It follows that the angles of an equilateral triangle (one having all sides equal) each measures 60o.
Let  be equilateral with each side having 2 units as
its length. Let M be the midpoint of side AC. See Figure 8.
Then  are congruent right triangles,
h2 + 12 = 22. |
Figure 8 |
side AM = 1, and
the length h of side MB satisfies 
and the three sides of this 30o, 60o, 90o triangle have lengths 1, 
2. If 
is any triangle with
then 
is similar to 
and its sides e, f, g must be proportional to 1,
2. We can write this as 
It follows that the lengths of the sides of a 30o, 60o, 90o triangle can be
written as 
Now let 
be an isosceles right triangle with 
c as the hypotenuse and k as the length of each of the other
two sides. Then 
and c2 = k2 + k2. It follows that 
and thus the sides of a 45o, 45o, 90o triangle
are of the form 
Before computers, when drafting was done by hand, every draftsman had several of these special triangles of various sizes in
his tool box. See figure 9a where 
is a 30o, 60o, 90o triangle and 
is a 45o, 45o, 90o triangle. With these, the
draftsman could construct many angles of various sizes. In figure 9b we see an example of how these two triangles and
addition of angles can be used to create a 75o angle, and in figure 9c we see an example of how one of the triangles and
subtraction is used to create an angle of 150o.
Figure 9a |
Figure 9b |
Figure 9c |
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