Propertics of triangle and circles connected with them
Triangle : The Area of a triangle is represented by the symbol triangle for any trianlge, the three sides are represented by a, b and c angles opposite these sides are represented by A,B,C respectively.
i) Where the measurements of three sides a,b,c are given
Area (A) = \(\sqrt {s(s - a)(s - b)(s - c)} \) \(\left\{ \begin{gathered}
s = \frac{{s + b + c}}{2}\,\,\, \hfill \\
s\,\,is\,\,\,semi\,\,perimeter \hfill \\
\end{gathered} \right.\)
ii) When base (b) and altitude or height (h) of the corresponding base are given.
\(Area(A) = \frac{1}{2} \times base \times altitude = \frac{1}{2}bh\)
iii) Area =\(1\over 2\) ab sinC =\(1\over 2\) bc sinA = \(1\over 2\)ca sinB
iv) Area =\(\frac{{abc}}{{4R}}\) where R is the circum radius of the triangle
v) Area = r.s, where ‘r’ is the inradius of the triangle and S, the semi perimeter.
vi) \(\Delta\)=2R2 sinA sinB sinC, where R is the circum radius of the triangle
Note : Generally, the first and the second formula most commonly used.
\(s = \frac{{a + b + c}}{2} = \frac{{sum\,\,of\,\,\,the\,\,length\,\,of\,the\,\,sides}}{2}\)
(a+b+c = perimeter of the triangle so, s is semi perimeter)