Oblique Triangle (Acute and Obtuse)

Written by Jerry Ratzlaff on . Posted in Plane Geometry

Oblique Trianglescalene triangle 5hscalene triangle 4acute triangle 1

  • An "oblique" triangle is any triangle that is not a right triangle.
  • An "acute" triangle is when all three angles of the triangle are less than right angles.
  • An "obtuse" triangle is when one of the three angles is greater than a right angle.
  • 3 edges
  • 3 vertexs
  • \(x\;+\;y\;+\;z\;=\;180°\).
  • Sidess:  \(a\),  \(b\),  \(c\)
  • Angles:  \(A\),  \(B\),  \(C\)
  • Height:  \(h_a\),  \(h_b\),  \(h_c\)obtuse triangle 4hobtuse triangle 3obtuse triangle 2
  • 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:  \(R\)
  • Area:  \(K\)
  • Perimeter:  \(P\)

 

scalene triangle 5m

scalene triangle 5t

 obtuse triangle 4mobtuse triangle 4t

 

 

 

 

 

 

 

Area of a Oblique Triangle

An oblique triangle includes an "acute" and "obtuse".

Area of a Oblique Triangle formula

\(K = \frac {hb} {2} \)

\(K = ab \frac {\sin y} {2} \)

Where:

\(K\) = area

\(b\) = side

Circumcircle of a Oblique Triangle

An oblique triangle includes an "acute" and "obtuse".

The radius of a circumcircle (outer) of a triangle if given all three sides \(( R )\).

Circumcircle of a Oblique Triangle formula

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

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

Where:

\(R\) = outcircle

\(a, b, c\) = side

\(s\) = semiperimeter

Height of a Oblique Triangle

An oblique triangle includes an "acute" and "obtuse".

Height of a Oblique Triangle formula

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

Where:

\(h\) = height

\(b\) = side

\(K\) = area

Inscribed Circle of a Oblique Triangle                                        

An oblique triangle includes an "acute" and "obtuse".

The radius of a inscribed circle (inner) of a triangle if given all three sides \(( r )\).

Inscribed Circle of a Oblique Triangle formula

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

Where:

\(r\) = incircle

\(a, b, c\) = side

Perimeter of a Oblique Triangle

An oblique triangle includes an "acute" and "obtuse".

Perimeter of a Oblique Triangle formula

\(P = a + b + c \)

Where:

\(P\) = perimeter

\(a, b, c\) = side

Semiperimeter of a Oblique Triangle

An oblique triangle includes an "acute" and "obtuse".

One half of the perimeter.

Semiperimeter of a Oblique Triangle formula

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

Where:

\(s\) = semiperimeter

\(a, b, c\) = side

Side of a Oblique Triangle

An oblique triangle includes an "acute" and "obtuse".

Side of a Oblique Triangle formula

\(a = P - b - c   \)

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

\(b = P - a - c   \)

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

\(c = P - a - b   \)

Where:

\(a, b, c\) = side

\(P\) = perimeter

\(K\) = area

Trig Function

  • 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