Right Triangle

Written by Jerry Ratzlaff on . Posted in Plane Geometry

Right Triangle

• One side of a right triangle is 90 °.
• The other two angles are unequal and no sides are equal.
• The hypotenuse of a right triangle is the longest side or the side opposite the right angle.
• 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
• Inradius of triangle:  $$r$$
• Outradius (circumcircle) of triangle:  $$R$$

Angle bisector of a Right Triangle

An angle bisector is a line that splits an angle into two equal angles.

Angle bisector of a Right Triangle formula

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

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

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

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

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

Where:

$$t_a, t_b, t_c$$ = angle bisector

$$A, B$$ = angle

$$a, b, c$$ = side

Area of a Right Triangle

Area of a Right Triangle formula

$$K= \frac {ab} {2}$$

Where:

$$K$$ = area

$$a, b$$ = side

Circumcircle of a Right Triangle

The radius of a circumcircle (outer) of a right triangle if given legs and hypotrnuse $$( R )$$.

Circumcircle of a Right Triangle formula

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

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

Where:

$$R$$ = outcircle

$$a, b$$ = side

$$H$$ = hypotenuse

Height of a Right Triangle

Height is the length of the two sides and the perpendicular height of the 90 degree angle.

Height of a Right Triangle formula

$$h_a = b$$

$$h_b = a$$

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

Where:

$$h_a, h_b, h_c$$ = hight

$$a, b, c$$ = side

Inscribed Circle of a Right Triangle

The radius of a inscribed circle (inner) of a right triangle if given legs and hypotrnuse $$( r )$$.

Inscribed Circle of a Right Triangle formula

$$r = \frac { ab } { a + b + c }$$

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

Where:

$$r$$ = incircle

$$a, b, c$$ = side

Median of a Right Triangle

Median is a line segment from a vertex (coiner point) to the midpoint of the opposite side.

Median of a Right Triangle formula

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

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

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

Where:

$$m_a, m_b, m_c$$ = median

$$a, b, c$$ = side

Perimeter of a Right Triangle

Perimeter of a Right Triangle formula

$$P = a + b + c$$

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

Where:

$$P$$ = perimeter

$$a, b, c$$ = side

Semiperimeter of a Right Triangle

One half of the perimeter.

Semiperimeter of a Right Triangle formula

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

Where:

$$s$$ = semiperimeter

$$a, b, c$$ = side

Trig Function

• 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 :  $$\; Area= ab \div 2$$