be equilateral with each side having 2 units as its length. Let M
be the midpoint of side AC. See Figure 7. Then
are congruent right triangles,
side AM = 1, and the length h of side MB satisfies
h2 + 12 = 22.
Figure 7
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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 7. Then
are congruent right triangles,
side AM = 1, and the length h of side MB satisfies h2 + 12 = 22. |
Figure 7 |
Thus 
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 
Figure 8 |
A parallelogram is a quadrilateral whose opposite sides are parallel.
It is a theorem that the quadrilateral ABCD is a parallelogram if
and only if  and 
have equal magnitudes and directions. See Figure 8. |
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