# Right Triangle

on . 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
• 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 :  $$\; a= c*sin \; A$$
• given A b :  $$\; a= b*tan \; A$$
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 :  $$\; Area= a\;b \div 2$$

Tags: Triangle