Plane Geometry, Plane Figures (Geometric Figures) - Triangles
     Right-angled Triangle - The Pythagorean Theorem
      The Pythagorean theorem
      Trigonometric functions of an acute angle defined in a right triangle
      Solving the right triangle
     Equilateral and Isosceles Triangle
      Equilateral triangle
      Isosceles triangle
Right-angled Triangle - The Pythagorean Theorem
The Pythagorean theorem
In any right triangle the area of the square whose side is the hypotenuse (the side of the triangle opposite the right angle) is equal to the sum of the areas of the squares on the other two sides (legs).
From the similarity of the triangles, ADC, BDC and ABC, and Thalesí theorem (an angle inscribed in a semicircle is a right angle) proved is Pythagorasí theorem:
In the figure below shown are two geometric proofs of Pythagoras' theorem which claims that the area of the square of the hypotenuse (the side opposite the right angle) is equal to the sum of areas of the squares of other two sides, i.e.,  c2 = a2 + b2.
First proof shows that the area of the biggest red square with the side a + b is equal to the sum of 
four equal right triangles and the square of the hypotenuse
c, therefore
                 (a + b)2 = 4 ∑ 1/2 ∑ ab + c2
       a2 + 2ab + b2 = 2ab + c
                   a2+ b2 = c
 
Second proof shows that the area of the square of the hypotenuse c is equal to the sum of the same four right triangles and the area of the small square with side a b, therefore
                 c2 = 4 ∑ 1/2 ∑ ab + (a - b)2
                 c2 = 2ab + a2 - 2ab + b2
                 c2 = a2+ b2
Trigonometric functions of an acute angle defined in a right triangle
Trigonometric functions of an acute angle are defined in a right triangle as a ratio of its sides.
 
Solving the right triangle
To solve a right triangle means to find all unknown sides and angles using its known parts.
While solving a right triangle we use Pythagorasí theorem and trigonometric functions of an acute angle depending which pair of its parts is given.
Note, right triangles are usually denoted as follows;  c stands for the hypotenuse,  a and b for the perpendicular sides called legs, and a and b for the angles opposite to a and b respectively. 
There are four basic cases that can occur, given
             a)  hypotenuse and angle,             c)  hypotenuse and leg,  
              b)  leg and angle,                        d)  two legs.
Equilateral and Isosceles Triangle
Equilateral triangle
                          perimeter  P = 3a
Isosceles triangle
 
perimeter  P = a + 2b
Geometry and use of trigonometry contents - A
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