Oblique Triangle (Acute and Obtuse)

Written by Jerry Ratzlaff on . Posted in Plane Geometry

  • acute triangle 1obtuse triangle 2A oblique triangle is a two-dimensional figure that is tilted at an angle, not horizontal or vertical.
  • An acute oblique triangle is a two-dimensional figure that has all three angles less than 90°.
  • An obtuse oblique triangle is a two-dimensional figure that has one of the three angles more than 90°.
  • 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
  • \(x\;+\;y\;+\;z\;=\;180°\).
  • Sidess:  \(a\),  \(b\),  \(c\)
  • 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

scalene triangle 5hobtuse triangle 4h

 

 

 

 

 

 

 

scalene triangle 5mscalene triangle 5tobtuse triangle 4mobtuse triangle 4t

 

 

 

 

 

 

 

obtuse triangle 2acute triangle 1Area of a Oblique Triangle formula

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

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

Where:

\(\large{ K }\) = area

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

 

obtuse triangle 3scalene triangle 4Circumcircle of a Oblique Triangle formula

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

\(\large{ R =  \frac  { a \; b \; c }   {   4 \;  \sqrt  {  s \; \left( s - a  \right)   \;   \left( s - b  \right)    \;    \left( s - c  \right)  }     }  }\)

Where:

\(\large{ R }\) = outcircle

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

\(\large{ s }\) = semiperimeter

obtuse triangle 2acute triangle 1Height of a Oblique Triangle formula

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

Where:

\(\large{ h }\) = height

\(\large{ K }\) = area

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

 

obtuse triangle 3Iscalene triangle 4nscribed Circle of a Oblique Triangle formula

\(\large{ r =   \sqrt  {   \frac  {  \left( s - a  \right)  \; \left( s - b  \right)  \; \left( s - c  \right)  }  { s }   }  }\)

Where:

\(\large{ r }\) = incircle

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

 

 

obtuse triangle 2acute triangle 1Perimeter of a Oblique Triangle formula

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

Where:

\(\large{P }\) = perimeter

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

 

 

obtuse triangle 2acute triangle 1Semiperimeter of a Oblique Triangle formula

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

Where:

\(\large{ s }\) = semiperimeter

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


 

 

obtuse triangle 2acute triangle 1Side of a Oblique Triangle formula

\(\large{ a = P - b - c   }\)

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

\(\large{ b = P - a - c   }\)

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

\(\large{ c = P - a - b   }\)

Where:

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

\(\large{ P }\) = perimeter

\(\large{ K }\) = area

Trig Functions

  • Find p
    • given a b c :  \(\; p= a+b+c \)
  • Find s
    • given a b c :  \(\; s= (a+b+c) \div 2 \)
  • Find d
    • given a b c s :  \(\; d= (b^2+c^2-a^2) \div 2b \)
  • Find e
    • given a b c s :  \(\; e= (a^2+b^2-c^2) \div 2b \)
  • Find h
    • given c A :  \(\; h= c*sin \;A \)
    • given a C :  \(\; h= a*sin \;C \)
  • Find Area
    • given a b c :  \(\; Area=\frac {1}{2}(bh) \)
    • given a b c :  \(\; Area=  \sqrt { s(s-a)(s-b)(s-c) } \)
    • given C a b :  \(\; Area= \frac{1}{2} ab* \sin \; c \)
  • Find A
    • given a b c s :  \(\; \sin \frac{1}{2} A=  \sqrt {(s-b)(s-c) \div bc } \)
    • given a b c s :  \(\; \cos \frac{1}{2} A= \sqrt {s(s-a) \div bc } \)
    • given a b c s :  \(\; \tan \frac{1}{2} A= \sqrt {(s-b)(s-c) \div s(s-a) } \)
    • given c h :  \(\;\ sin \; A= \frac {h}{c} \)
    • given B a b :  \(\;\ sin \; A= a* sin\; B \div b \)
    • given B a c :  \(\; A= \frac{1}{2}(A+C)+\frac{1}{2}(A-C) \)
    • given C a b :  \(\; A= \frac{1}{2}(A+B)+\frac{1}{2}(A-B) \)
    • given C a c :  \(\;\ sin \; A= a* sin \; C \div c \)
  • Find B
    • given a b c s :  \(\; \sin \frac{1}{2} B= \sqrt {(s-a)(s-c) \div ac} \)
    • given a b c s :  \(\; \cos \frac{1}{2} B= \sqrt {s(s-b) \div ac} \)
    • given a b c s :  \(\; \tan \frac{1}{2} B= \sqrt {(s-a)(s-c) \div s(s-b)} \)
    • given a h :  \(\;\ sin \; B= \frac {h}{a} \)
    • given A a b :  \(\;\ sin B= b* sin \; A \div a \)
    • given A b c :  \(\;\ B= \frac{1}{2}(B+C)+\frac{1}{2}(B-C) \)
    • given C a b :  \(\; B= \frac{1}{2}(A+B)-\frac{1}{2}(A-B) \)
    • given C a c :  \(\;\ sin \; B= b* sin \; C \div c \)
  • Find C
    • given a b c s :  \(\; \sin \frac{1}{2} C= \sqrt {(s-a)(s-b) \div ab} \)
    • given a b c s :  \(\; \cos \frac{1}{2} C= \sqrt {s(s-c) \div ab} \)
    • given a b c s :  \(\; \tan \frac{1}{2} C= \sqrt {(s-a)(s-b) \div s(s-c) } \)
    • given A a c :  \(\;\ sin \; C= c* sin \; A \div a \)
    • given A b c :  \(\; C= \frac{1}{2}(B+C)-\frac{1}{2}(B-C) \)
    • given B a c :  \(\; C= \frac{1}{2}(A+C)-\frac{1}{2}(A-C) \)
    • given B b c :  \(\;\ sin \; C= c* sin \; B \div c \)
  • Find a
    • given A B b :  \(\; a= b* \sin \; A \div \sin \; B \)
    • given A B c :  \(\; a= c* \sin \; A \div \sin (A+B) \)
    • given A C b :  \(\; a= b* \sin \; A \div \sin (A+C) \)
    • given A C c :  \(\; a= c* \sin \; A \div \sin \; C \)
    • given B C b :  \(\; a= b* \sin (A+C) \div \sin \; B \)
    • given B C c :  \(\; a= c* \sin (A+C) \div \sin \; C \)
    • given A b c :  \(\; a= \sqrt {b^2+c^2-2bc * \cos \; A} \)
  • Find b
    • given A B a :  \(\; b= a* \sin \; B \div \sin \; A \)
    • given A B c :  \(\; b= c* \sin \; B \div \sin (A+B) \)
    • given A C a :  \(\; b= a* \sin (A+C) \div \sin \; A \)
    • given A C c :  \(\; b= c* \sin (A+C) \div \sin \; C \)
    • given B C a :  \(\; b= a* \sin \; B \div \sin (B+C) \)
    • given B C c :  \(\; b= c* \sin \; B \div \sin \; C \)
    • given B a c :  \(\; b= \sqrt {a^2+c^2-2ac * \cos B} \)
  • Find c
    • given A B a :  \(\; c= a* \sin (A+B) \div \sin \; A \)
    • given A B b :  \(\; c= b* \sin (A+B) \div \sin \; B \)
    • given A C a :  \(\; c= a* \sin \; C \div \sin \; A \)
    • given A C b :  \(\; c= b* \sin \; C \div \sin (A+C) \)
    • given B C a :  \(\; c= a* \sin \; C \div \sin (B+C) \)
    • given B C b :  \(\; c= b* \sin \; C \div \sin \; B \)
    • given C a b :  \(\; c= \sqrt {a^2+b^2-2ab * \cos \; C} \)

     

Tags: Equations for Triangle