Special Right Triangles

THE 45-45-90 TRIANGLE

This is a triangle whose three angles respectively measure 45°, 45°, and 90°. The sides are in the ratio

:

1:1:\sqrt{2}.

A simple proof. Suppose you have such a triangle with legs ''a'' and ''b'' and Hypotenuse ''c''. Suppose that ''a'' = 1. Since two angles measure 45°, this is an Isosceles Triangle and we have ''b'' = 1. The fact that

c=\sqrt{2}

follows immediately from the Pythagorean Theorem .

THE 30-60-90 TRIANGLE

This is a triangle whose three angles respectively measure 30°, 60°, and 90°. The sides are in the ratio

:

1:\sqrt{3}:2.

The proof of this fact is obvious using Trigonometry . Although the Geometric is less apparent, it is equally trivial:

:Draw an equilateral triangle ''ABC'' with side length ''2'' and with point ''D'' as the midpoint of segment ''BC''. Draw an altitude line from ''A'' to ''D''. Then ''ABD'' is a 30-60-90 triangle with hypotenuse of length ''2'', and base ''BD'' of length ''1''.

:The fact that the remaining leg ''AD'' has length

\sqrt{3}

follows immediately from the Pythagorean Theorem .

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Special Right Triangles