JEE Main & Advanced Mathematics Rectangular Cartesian Coordinates Area of Some Geometrical Figures

(1) Area of a triangle : The area of a triangle ABC with vertices \[A({{x}_{1}},{{y}_{1}}),\,\,B\text{ }({{x}_{2}},{{y}_{2}})\] and \[C({{x}_{3}},{{y}_{3}})\]. The area of triangle ABC is denoted by \['\Delta '\]and is given as

\[\Delta =\frac{1}{2}\left| \begin{matrix} {{x}_{1}} & {{y}_{1}} & 1  \\    {{x}_{2}} & {{y}_{2}} & 1  \\ {{x}_{3}} & {{y}_{3}} & 1  \\ \end{matrix} \right|\]\[=\frac{1}{2}\left| \text{ }({{x}_{1}}({{y}_{2}}-{{y}_{3}})+{{x}_{2}}({{y}_{3}}-{{y}_{1}})+{{x}_{3}}({{y}_{1}}-{{y}_{2}})\text{ } \right|\]

In equilateral triangle

(i) Having sides a, area is \[\frac{\sqrt{3}}{4}{{a}^{2}}\].

(ii) Having length of perpendicular as 'p' area is \[\frac{({{p}^{2}})}{\sqrt{3}}\] .

(2) Collinear points : Three points \[A({{x}_{1}},{{y}_{1}}),\,\,B({{x}_{2}},{{y}_{2}}),\,C({{x}_{3}},{{y}_{3}})\] are collinear.  If area of triangle is zero, then

(i)  \[\Delta =0\]  \[\Rightarrow \]  \[\frac{1}{2}\left| \begin{matrix} {{x}_{1}} & {{y}_{1}} & 1  \\ {{x}_{2}} & {{y}_{2}} & 1  \\ {{x}_{3}} & {{y}_{3}} & 1  \\ \end{matrix} \right|=0\] \[\Rightarrow \] \[\left| \begin{matrix} {{x}_{1}} & {{y}_{1}} & 1  \\ {{x}_{2}} & {{y}_{2}} & 1  \\ {{x}_{3}} & {{y}_{3}} & 1  \\ \end{matrix} \right|=0\]

(ii) \[AB+BC=AC\] or \[AC+BC=AB\] or \[AC+AB=BC\]

(3) Area of a quadrilateral : If \[({{x}_{1}},{{y}_{1}}),\,({{x}_{2}},{{y}_{2}}),\,\,({{x}_{3}},{{y}_{3}})\] and \[({{x}_{4}},{{y}_{4}})\] are vertices of a quadrilateral, then its area

\[=\frac{1}{2}[({{x}_{1}}{{y}_{2}}-{{x}_{2}}{{y}_{1}})+({{x}_{2}}{{y}_{3}}-{{x}_{3}}{{y}_{2}})+({{x}_{3}}{{y}_{4}}-{{x}_{4}}{{y}_{3}})+({{x}_{4}}{{y}_{1}}-{{x}_{1}}{{y}_{4}})]\]

(4) Area of polygon : The area of polygon whose vertices are \[({{x}_{1}},{{y}_{1}}),({{x}_{2}},{{y}_{2}}),({{x}_{3}},{{y}_{3}}),....({{x}_{n,}}{{y}_{n}})\] is

\[=\,\frac{1}{2}|\{({{x}_{1}}{{y}_{2}}-{{x}_{2}}{{y}_{1}})+({{x}_{2}}{{y}_{3}}-{{x}_{3}}{{y}_{2}})+....+({{x}_{n}}{{y}_{1}}-{{x}_{1}}{{y}_{n}})\}|\]                

Or   Stair method : Repeat first co-ordinates one time in last for down arrow use positive sign and for up arrow use negative sign.

\[\therefore \] Area of polygon

\[=\frac{1}{2}|\{({{x}_{1}}{{y}_{2}}+{{x}_{2}}{{y}_{3}}+....+{{x}_{n}}{{y}_{1}})-({{y}_{1}}{{x}_{2}}+{{y}_{2}}{{x}_{3}}+....+{{y}_{n}}{{x}_{1}})\}|\]