Area of Some Geometrical Figures
Category : JEE Main & Advanced
(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
\[=\frac{1}{2}|\{({{x}_{1}}{{y}_{2}}+{{x}_{2}}{{y}_{3}}+....+{{x}_{n}}{{y}_{1}})-({{y}_{1}}{{x}_{2}}+{{y}_{2}}{{x}_{3}}+....+{{y}_{n}}{{x}_{1}})\}|\]
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