
Plane Geometry, Plane Figures (Geometric
Figures)  Triangles 


Rightangled
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






Rightangled
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
Thales’ theorem (an angle inscribed in a semicircle is a right angle)
proved is Pythagoras’ 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., c^{2}
= a^{2}
+ b^{2}. 
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
+ c^{2} 
a^{2
}+
2ab
+ b^{2}
= 2ab
+ c^{2 } 
a^{2}+
b^{2}
= c^{2 } 


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 
c^{2} = 4
· 1/2 · ab
+ (a

b)^{2} 
c^{2} = 2ab
+ a^{2}

2ab
+ b^{2} 
c^{2} =
a^{2}+
b^{2} 





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
Pythagoras’ 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



Isosceles
triangle










Geometry
and use of trigonometry contents  A 



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