# Right Triangle

Written by Jerry Ratzlaff on . Posted in Plane Geometry

• A two-dimensional figure that 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.
• 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.
• Median of a right triangle is a line segment from a vertex (coiner point) to the midpoint of the opposite side.
• Circumcircle (R) is a circle that passes through all the vertices of a two-dimensional figure.
• Inscribed circle (r) is the largest circle possible that can fit on the inside of a two-dimensional figure.
• Semiperimeter (s) is one half of the perimeter.
• 3 edges
• 3 vertexs
• $$a$$ = opposite leg
• $$b$$ = adjacent leg
• $$c$$ = hypotenuse
• Angles:  $$A$$,  $$B$$,  $$C$$
• Area:  $$K$$
• Perimeter:  $$P$$
• 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
• Semi-perimeter:  $$s$$  -  One half of the perimeter

### Angle bisector of a Right Triangle formula

$$\large{ t_a = 2\;b\;c \; cos \; \frac { \frac {A}{2} }{ b + c } }$$

$$\large{ t_a = \sqrt { b\;c \; \frac { 1 - a^2 } { \left( b + c \right)^2 } } }$$

$$\large{ t_b = 2\;a\;c \; cos \; \frac { \frac {B}{2} }{ a + c } }$$

$$\large{ t_b = \sqrt { a\;c \; \frac { 1 - b^2 } { \left( a + c \right)^2 } } }$$

$$\large{ t_c = a\;b \; \sqrt { \frac { 2 }{ a + b } } }$$

Where:

$$\large{ t_a, t_b, t_c }$$ = angle bisector

$$\large{ A, B }$$ = angle

$$\large{ a, b, c }$$ = side

### Area of a Right Triangle formula

$$\large{ K= \frac {a\;b} {2} }$$

Where:

$$\large{ K }$$ = area

$$\large{ a, b }$$ = side

### Circumcircle of a Right Triangle formula

$$\large{ R = \frac { 1 } { 2 } \; \sqrt { a^2 + b^2 } }$$

$$\large{ R = \frac { H } { 2 } }$$

Where:

$$\large{ R }$$ = outcircle

$$\large{ a, b }$$ = side

$$\large{ H }$$ = hypotenuse

### Height of a Right Triangle formula

$$\large{ h_a = b }$$

$$\large{ h_b = a }$$

$$\large{ h_c = \frac {a \;b} {c} }$$

Where:

$$\large{ h_a, h_b, h_c }$$ = hight

$$\large{ a, b, c }$$ = side

### Inscribed Circle of a Right Triangle formula

$$\large{ r = \frac { a\;b } { a + b + c } }$$

$$\large{ r = \frac { a + b - c } { 2 } }$$

Where:

$$\large{ r }$$ = incircle

$$\large{ a, b, c }$$ = side

### Median of a Right Triangle formula

$$\large{ m_a = \sqrt { \frac { 4\;b^2 + a^2 }{ 2 } } }$$

$$\large{ m_b = \sqrt { \frac { 4\;a^2 + b^2 }{ 2 } } }$$

$$\large{ m_c = \frac {c} {2} }$$

Where:

$$\large{ m_a, m_b, m_c }$$ = median

$$\large{ a, b, c }$$ = side

### Perimeter of a Right Triangle formula

$$\large{ P = a + b + c }$$

$$\large{ P = a + b + \sqrt {a^2 + b^2 } }$$

Where:

$$\large{ P }$$ = perimeter

$$\large{ a, b, c }$$ = side

### Semiperimeter of a Right Triangle formula

$$\large{ s = \frac { a + b + c } { 2 } }$$

Where:

$$\large{ s }$$ = semiperimeter

$$\large{ a, b, c }$$ = side

## 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 :  $$\; Area= a\;b \div 2$$