Right Triangle

on 29 January 2016. Posted in Plane Geometry

Right triangle (a two-dimensional figure) has one side a right 90° interior angle.
The other two angles are unequal and no sides are equal.
Angle bisector of a right triangle is a line that splits an angle into two equal angles.
Circumcircle is a circle that passes through all the vertices of a two-dimensional figure.
Height of a right triangle is the length of the two sides and the perpendicular height of the 90 degree angle.
Hypotenuse of a right triangle is the longest side or the side opposite the right angle.
Inscribed circle is the largest circle possible that can fit on the inside of a two-dimensional figure.
Median of a right triangle is a line segment from a vertex (coiner point) to the midpoint of the opposite side.
Semiperimeter is one half of the perimeter.
3 edges
3 vertexs
a = opposite leg
b = adjacent leg
c = hypotenuse
Angles: ∠A, ∠B, ∠C
Height: \(h_a\), \(h_b\), \(h_c\)
Median: \(m_a\), \(m_b\), \(m_c\) - A line segment from a vertex (coiner point) to the midpoint of the opposite side
Angle bisectors: \(t_a\), \(t_b\), \(t_c\) - A line that splits an angle into two equal angles

Angle bisector of a Right Triangle formulas
\(\large{ t_a = 2\;b\;c \; cos \; \frac { \frac {A}{2} }{ b \;+\; c } }\) \(\large{ t_a = \sqrt { b\;c \; \frac { 1 \;-\; a^2 } { \left( b \;+\; c \right)^2 } } }\) \(\large{ t_b = 2\;a\;c \; cos \; \frac { \frac {B}{2} }{ a \;+\; c } }\) \(\large{ t_b = \sqrt { a\;c \; \frac { 1 \;-\; b^2 } { \left( a \;+\; c \right)^2 } } }\) \(\large{ t_c = a\;b \; \sqrt { \frac { 2 }{ a \;+\; b } } }\)
Symbol	English	Metric
\(\large{ t_a, t_b, t_c }\) = angle bisector	\(\large{ in}\)	\(\large{ mm}\)
\(\large{ A, B, C }\) = angle	\(\large{deg}\)	\(\large{rad}\)
\(\large{ a, b, c }\) = edge	\(\large{ in}\)	\(\large{ mm }\)

Area of a Right Triangle formula
\(\large{ A_{area} = \frac {a\;b} {2} }\)
Symbol	English	Metric
\(\large{ A_{area} }\) = area	\(\large{ in}\)	\(\large{ mm}\)
\(\large{ a, b, c }\) = edge	\(\large{ in}\)	\(\large{ mm }\)

Circumcircle of a Right Triangle formulas
\(\large{ R = \frac { 1 } { 2 } \; \sqrt { a^2 + b^2 } }\) \(\large{ R = \frac { H } { 2 } }\)
Symbol	English	Metric
\(\large{ R }\) = outcircle	\(\large{ in}\)	\(\large{ mm}\)
\(\large{ a, b, c }\) = edge	\(\large{ in}\)	\(\large{ mm }\)
\(\large{ H }\) = hypotenuse	\(\large{ in}\)	\(\large{ mm }\)

Height of a Right Triangle formulas
\(\large{ h_a = b }\) \(\large{ h_b = a }\) \(\large{ h_c = \frac {a \;b} {c} }\)
Symbol	English	Metric
\(\large{ h_a, h_b, h_c }\) = hight	\(\large{ in}\)	\(\large{ mm}\)
\(\large{ a, b, c }\) = edge	\(\large{ in}\)	\(\large{ mm }\)

Hypotenuse of a Right Triangle formula
\(\large{ c = \sqrt{a^2+b^2} }\)
Symbol	English	Metric
\(\large{ c }\) = hypotenuse (H)	\(\large{ in}\)	\(\large{ mm}\)
\(\large{ a, b, c }\) = edge	\(\large{ in}\)	\(\large{ mm }\)

Inscribed Circle of a Right Triangle formulas
\(\large{ r = \frac { a\;b } { a \;+\; b \;+\; c } }\) \(\large{ r = \frac { a \;+\; b \;-\; c } { 2 } }\)
Symbol	English	Metric
\(\large{ r }\) = incircle	\(\large{ in}\)	\(\large{ mm}\)
\(\large{ a, b, c }\) = edge	\(\large{ in}\)	\(\large{ mm }\)

Median of a Right Triangle formulas
\(\large{ m_a = \sqrt { \frac { 4\;b^2 \;+\; a^2 }{ 2 } } }\) \(\large{ m_b = \sqrt { \frac { 4\;a^2 \;+\; b^2 }{ 2 } } }\) \(\large{ m_c = \frac {c} {2} }\)
Symbol	English	Metric
\(\large{ m_a, m_b, m_c }\) = median	\(\large{ in}\)	\(\large{ mm}\)
\(\large{ a, b, c }\) = edge	\(\large{ in}\)	\(\large{ mm }\)

Perimeter of a Right Triangle formulas
\(\large{ P = a + b + c }\) \(\large{ P = a + b + \sqrt {a^2 + b^2 } }\)
Symbol	English	Metric
\(\large{ P }\) = perimeter	\(\large{ in}\)	\(\large{ mm}\)
\(\large{ a, b, c }\) = edge	\(\large{ in}\)	\(\large{ mm }\)

Semiperimeter of a Right Triangle formula
\(\large{ s = \frac { a \;+\; b \;+\; c } { 2 } }\)
Symbol	English	Metric
\(\large{ s }\) = semiperimeter	\(\large{ in}\)	\(\large{ mm}\)
\(\large{ a, b, c }\) = edge	\(\large{ in}\)	\(\large{ mm }\)

Trig Functions
Find A given a c : \(\; sin \; A= a \div c \) given b c : \(\; cos \; A= b \div c \) given a b : \(\; tan \; A= a \div b \)
Find B given a c : \(\; sin \; B= a \div c \) given b c : \(\; cos \; B= b \div c \) given a b : \(\; tan \; B= b \div a \)
Find a given A c : \(\; a= csin \; A \) given A b : \(\; a= btan \; A \)
Find b given A c : \(\; b= c*cos \; A \) given A a : \(\; b= a \div tan \; A \)
Find c given A a : \(\; c= a \div sin \; A \) given A b : \(\; c= b \div cos \; A \) given a b : \(\; c= \sqrt { a^2+b^2 } \)
Find Area given a b : \(\; Area= a\;b \div 2 \)

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Angle bisector of a Right Triangle formulas