Right Isosceles Triangle

Written by Jerry Ratzlaff on . Posted in Plane Geometry

right isosceles triangle 2

  • Right isosceles triangle (a two-dimensional figure) has one side a right 90° interior angle and the other two angles are 45°.
  • Angle bisector of a right isosceles 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.
  • Hypotenuse of a right isosceles triangle is the longest side or the side opposite the right angle.
  • Inscribed circle is the Iargest circle possible that can fit on the inside of a two-dimensional figure.
  • Median of a right isosceles triangle is a line segment from a vertex (coiner point) to the midpoint of the opposite side.
  • Semiperimeter is one half of the perimeter.
  • Side of a right triangle is one half of the perimeter.
  • Two sides are congruent.
  • 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 (corner 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

right isosceles triangle 4tAngle bisector of a Right Isosceles Triangle formula

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

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

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

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

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

Where:

\(\large{ t_a, t_b, t_c }\) = angle bisector

\(\large{ A, B }\) = angle

\(\large{ a, b, c }\) = edge

Area of a Right Isosceles Triangle formula

\(\large{ A_{area} = \frac {h\;b} {2} }\)

\(\large{ A_{area} = \frac {1} {2}\; b\;h }\)

\(\large{ A_{area} = a\;b\; \frac {\sin \;y} {2} }\)

Where:

\(\large{ A_{area} }\) = area

\(\large{ a, b, c }\) = edge

\(\large{ h }\) = height

right isosceles triangle 3Circumcircle of a Right sosceles Triangle formula

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

\(\large{ R =  \frac  { H } { 2 }   }\)

Where:

\(\large{ R }\) = outcircle

\(\large{ a, b, c }\) = edge

\(\large{ H }\) = hypotenuse

right isosceles triangle 4hHeight of a Right Isosceles Triangle formula

\(\large{ h_c = 2\; \frac {A_{area}}{b} }\)

Where:

\(\large{ h^c }\) = height

\(\large{ A_{area} }\) = area

\(\large{ a, b, c }\) = edge

 

right isosceles triangle 3Inscribed Circle of a Right Isosceles Triangle formula

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

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

Where:

\(\large{ r }\) = incircle

\(\large{ a, b, c }\) = edge

 

right isosceles triangle 4mMedian of a Right Isosceles 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

\(\large{ a, b, c }\) = edge

Perimeter of a Right Isosceles Triangle formula

\(\large{ P = a + b + c }\)

Where:

\(\large{ P }\) = perimeter

\(\large{ a, b, c }\) = edge

Side of a Right Isosceles Triangle formula

\(\large{ a = P - b - c   }\)

\(\large{ a = 2\; \frac {A_{area}} {b\;\sin y} }\)

\(\large{ b = P - a - c   }\)

\(\large{ b = 2\; \frac {A_{area}}{h} }\)

\(\large{ c = P - a - b   }\)

Where:

\(\large{ a, b, c }\) = edge

\(\large{ A_{area} }\) = area

\(\large{ P }\) = perimeter

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: Equations for Triangle