Special Right Triangles
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Key Questions

There are 2 types of special right triangles.
Type 1. Triangle that is half of a equilateral triangle. Its 3 angle measures are: 30, 60 and 90 deg. Its side measures are : a, a/2; and (a*sqr.3)/2.
Type 2. Triangle that has its side measures in the ratio of 3:4:5. The proof is given by the Pythagor theorem: c^2 = b^2 + a^2.
Use of special right triangles.
In the old time, people use the special right triangles with sides ratio 3:4:5 to figure out, in the field, a right angle or a rectangular, or square, shape.
Now, students just use the properties of special right triangle to find, by computing, the unknown sides or angles. 
The basic properties are: An interior angle of
#45^@# , an exterior angle of#135^@# , an opposite angleof#45^@# , a right angle (#90^@# ), opposite side length:#sqrt2/2# , and adjacent side length of#sqrt2/2# , and a hypotenuse of 1. (side lengths for a standard triangle inside of the Unit Circle)Explanation
All standard right triangles inside of the unit circle have a hypotenouse of 1. This is because the unit circle is defined as a circle with a radius of 1. Thus we create Unit Circle Triangles by picking a radius, and drawing its vector components.
In this case, we know we have a triangle with a primary angle of
#45^@# and, as mentioned before, a hypotneouse of 1. As such we can use#sintheta# and#costheta# to determine the other two lengths.#sintheta=o/h>hsintheta=o>1*sin(45^@)=sqrt2/2# #costheta=a/h>hcostheta=a>1*cos(45^@)=sqrt2/2# 
#mathbf{30^circ""60^circ""90^circ}# TriangleThe ratios of three sides of a
#30^circ""60^circ""90^circ# triangle are:#1:sqrt{3}:2#
I hope that this was helpful.

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