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Geometry/Chapter 20

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Some right triangles are particularly easy to solve; they are known as special right triangles. There are two kinds: 45-45-90 triangles and 30-60-90 triangles, named by their angles in degrees. All the triangles in each group are similar to each other.

Section 20.1 - 45-45-90 Triangles

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45-45-90 triangles are the only triangles that are both isosceles and right. Their legs are of the same length, and their hypotenuse is the square root of two times the length of a leg. For example, if ABC is an isosceles right triangle with A being the right angle and AB having length 3, then AC also has length 3 and BC has length 3*sqrt(2), or 4.242... The relationship can easily be found by the Pythagorean theorem: 1^2+1^2=2, because a^2+b^2=c^2. Drawing the altitude to the hypotenuse of an isosceles right triangle splits into two smaller isosceles right triangles, each similar to the original with a side length ratio of sqrt(2).

Section 20.2 - 30-60-90 Triangles

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30-60-90 triangles have a different length ratio--1:square root of three:2. This is confirmed by the Pythagorean theorem as well: 1^2+3=2^2. Clearly, the smallest side is opposite the smallest angle, so for example, in triangle ABC, with angles A, B, and C having measure 30, 60, and 90 degrees respectively and AB having length 1, BC will have length 1/2 and AC will have length sqrt(3)/2, or 0.866...

Of course, these triangles could be solved by trigonometry, but these ratios provide a shortcut. In fact, they help us remember the most important trigonometric values in the 0-to-90 degree range: 

sin(0)=0   sin(30)=1/2        sin(45)=sqrt(2)/2, or 1/sqrt(2)  sin(60)=sqrt(3)/2   sin(90)=1
cos(0)=1   cos(30)=sqrt(3)/2  cos(45)=sqrt(2)/2, or 1/sqrt(2)  cos(60)=1/2         cos(90)=0
tan(0)=0   tan(30)=sqrt(3)/3  tan(45)=1                        tan(60)=sqrt(3)     tan(90) is not defined.

Note that sine divided by cosine equals tangent, and also that
sin(90-x)=cos x, cos(90-x)=sin x, and tan(90-x)=1/tan x.

Exercises

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1. A triangle ABC is right and has AB=BC=5.
Find AC and the angle measures.
2. A triangle XYZ has X=30 degrees, Y=90 degrees, and XY=6 meters.
Find the measure of Z and the lengths of XZ and YZ.
3. A right-triangular yard is to be fenced off behind a house 10 meters across. The house is to be the hypotenuse of the yard, and one of the fence-to-house angles is to be twice as large as the other.
Find the amount of fence needed.

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