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Oblique Triangle (Acute and Obtuse)

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  • acute triangle 1obtuse triangle 2Oblique triangle (a two-dimensional figure) is tilted at an angle, not horizontal or vertical.
  • Acute oblique triangle (a two-dimensional figure) has all three angles less than 90°.
  • Circumcircle is a circle that passes through all the vertices of a two-dimensional figure.
  • Inscribed circle is the largest circle possible that can fit on the inside of a two-dimensional figure.
  • Obtuse oblique triangle (a two-dimensional figure) has one of the three angles more than 90°.
  • Semiperimeter is one half of the perimeter.
  • x + y + z = 180°
  • 3 edges
  • 3 vertexs
  • 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

Oblique Triangle (Acute and Obtuse) Index

scalene triangle 5hobtuse triangle 4h

obtuse triangle angle bisector 1

obtuse triangle circ 1

 

 

 

 

 

 

 

scalene triangle 5mscalene triangle 5tobtuse triangle 4t

scalene triangle 4

 

 

 

 

 

 

 

 

area of an Oblique triangle formulas

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

\(\large{ A_{area} = a\;b\; \frac {\sin y} {2} }\) 
Symbol English Metric
\(\large{ A_{area} }\) = area \(\large{ in^2 }\)   \(\large{ mm^2 }\)
\(\large{ a, b, c }\) = edge \(\large{ in }\) \(\large{ mm }\)

 

Circumcircle of an Oblique triangle formulas

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

\(\large{ R =  \frac  { a \; b \; c }   {   4 \;  \sqrt  {  s \; \left( s \;-\; a  \right)   \;   \left( s \;-\; b  \right)    \;    \left( s \;-\; c  \right)  }     }  }\) 
Symbol English Metric
\(\large{ R }\) = outcircle \(\large{ in }\)   \(\large{ mm }\)
\(\large{ a, b, c }\) = edge \(\large{ in }\) \(\large{ mm }\)
\(\large{ s }\) = semiperimeter \(\large{ in }\) \(\large{ mm }\)

 

height of an Oblique triangle formula
\(\large{ h = 2\; \frac {A_{area}}{b} }\) 
Symbol English Metric
\(\large{ h }\) = height \(\large{ in }\) \(\large{ mm }\)
\(\large{ A_{area} }\) = area \(\large{ in^2 }\)   \(\large{ mm^2 }\)
\(\large{ a, b, c }\) = edge \(\large{ in }\) \(\large{ mm }\)

 

inscribed circle of an Oblique triangle formula
\(\large{ r =   \sqrt  {   \frac  {  \left( s \;-\; a  \right)  \; \left( s \;-\; b  \right)  \; \left( s \;-\; c  \right)  }  { s }   }  }\) 
Symbol English Metric
\(\large{ r }\) = incircle \(\large{ in }\)   \(\large{ mm }\)
\(\large{ a, b, c }\) = edge \(\large{ in }\) \(\large{ mm }\)

 

perimeter of an Oblique triangle formula
\(\large{P = a + b + c }\) 
Symbol English Metric
\(\large{P }\) = perimeter \(\large{ in }\)   \(\large{ mm }\)
\(\large{ a, b, c }\) = edge \(\large{ in }\) \(\large{ mm }\)

 

semiperimeter of an Oblique triangle formula
\(\large{ s =   \frac  { a \;+\; b \;+\; c }  { 2  }   }\) 
Symbol English Metric
\(\large{ s }\) = semiperimeter \(\large{ in }\)   \(\large{ mm }\)
\(\large{ a, b, c }\) = edge \(\large{ in }\) \(\large{ mm }\)

 

SIDE of an Oblique triangle formulas

\(\large{ a = P - b - c   }\) 

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

\(\large{ b = P - a - c   }\) 

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

\(\large{ c = P - a - b   }\)
Symbol English Metric
\(\large{ a, b, c }\) = edge \(\large{ in }\) \(\large{ mm }\)
\(\large{ A_{area} }\) = area \(\large{ in^2 }\)   \(\large{ mm^2 }\)
\(\large{ P }\) = perimeter \(\large{ in }\) \(\large{ mm }\)

  

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 :  \(\; A_{area}=\frac {1}{2}(bh) \)
  • given a b c :  \(\; A_{area}=  \sqrt { s(s-a)(s-b)(s-c) } \)
  • given C a b :  \(\; A_{area}= \frac{1}{2} ab* \sin \; c \)
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 :  \(\; c= \sqrt {a^2+b^2-2ab * \cos \; C} \)

 

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Tags: Triangle

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