Determine Area of an Irregular Plane Figure using Geogebra Classic 5

\[1.\ \text{To mark the required number of points}\ \hspace{19cm}\\ \text{on the boundary of the figure.}\ \hspace{10cm}\]

\[2.\ \text{To draw the boundary of the figure}\ \hspace{19cm}\\ \text{by joining the points}\ \hspace{10cm}\]

\[3.\ \text{To divide the figure into trapeziums using}\ \hspace{19cm}\\ \text{the points on the boundary.}\ \hspace{10cm}\]

\[4.\ \text{To calculate the approximate are of the figure.}\ \hspace{18cm}\]

\[\color{green}{Step\ 1:}\ \text{Open Geogebra classic 5 (by double clicking on the icon)}\ \hspace{18cm}\]

\[\color{green}{Step\ 2:}\ \text{Impost the given image of the given irregular}\ \hspace{18cm}\\ \text{plane figure by using the Image tool.}\ \hspace{10cm}\]

\[\color{green}{Step\ 3:}\ \hspace{25cm}\]

\[i.\ \hspace{1cm}\ \text{Two points A and B are provided to the set}\ \hspace{15cm}\\ \text{the coordinates of the image.}\ \hspace{13cm}\]

\[ii.\ \hspace{1cm}\ \text{Without changing the dimension of the image, drag the}\ \hspace{15cm}\\ \text{image and fix the points A and B on the graphics view.}\ \hspace{13cm}\]

\[iii.\ \hspace{1cm}\ \text{Right click on the image, go to Object Properties}\ \hspace{15cm}\\ \text{and set a Background image.}\ \hspace{13cm}\]

\[\color{green}{Step\ 4:}\ \hspace{1cm}\ \text{To mark the vertices of the irregular}\ \hspace{18cm}\\ \text{polygon using point tool.}\ \hspace{13cm}\]

\[\color{green}{Step\ 5:}\ \hspace{25cm}\]

\[i.\ \hspace{1cm}\ \text{For each point representing a vertex, mark another point on the}\ \hspace{15cm}\\ \text{boundary of the polygon just vertically above.}\ \hspace{13cm}\\ \text{or below the point if possible}\ \hspace{12cm}\]

\[ii.\ \hspace{1cm}\ \text{After marking the point, adjust the x – coordinate of the newly}\ \hspace{15cm}\\ \text{marked point in the input bar to match with the}\ \hspace{13cm}\\ \text{x-coordinate of the vertex point}\ \hspace{12cm}\]

\[\color{green}{Step\ 6:}\ \text{Let the vertices are marked with the points C,D,E,F,G and}\ \hspace{18cm}\\ \text{corresponding vertical opposite points on the boundary of the}\ \hspace{14cm}\\ \text{polygon corresponding to D,F and G are marked with}\ \hspace{14cm}\\ \text{the points H, I and J respectively}\ \hspace{10cm}\]

\[\color{green}{Note:}\ \text{Record x and y coordinates of the points C,D,E,F,G, H, I}\ \hspace{18cm}\\ \text{and J by looking them in the input bar}\ \hspace{10cm}\]

\[\color{green}{Step\ 7:}\ \text{Now the entire irregular polygon can be divided into triagnles}\ \hspace{18cm}\\ \text{and trapeziums using the vertices and their vertically}\ \hspace{14cm}\\ \text{opposite points on the boundary of the polygon}\ \hspace{10cm}\]

\[\color{green}{Note:}\ \text{In this problem the polygon is divided into triangle CCJ, trapezium jGHD,}\ \hspace{15cm}\\ \text{trapezium HFID and triangle FEI}\ \hspace{10cm}\]

\[\color{green}{Step\ 8:}\ \text{Calculate the area of each subdivision of the polygon as following,}\ \hspace{15cm}\\ \text{for illustration, we take the trapezium GHDJ}\ \hspace{14cm}\]

\[\color{green}{Step\ 9:}\ \text{Join the points J and D by a straight line}\ \hspace{18cm}\\ \text{using the command l1:Line(J,D)}\ \hspace{14cm}\]

\[\color{green}{Step\ 10:}\ \text{Join the points G and H by a straight line}\ \hspace{18cm}\\ \text{using the command l2:Line(G,H)}\ \hspace{14cm}\]

\[\color{green}{Step\ 11:}\ \text{Calculate the Area of the trapezium CADJ }\ \hspace{18cm}\\ \text{using the command A2=Integral between (l1,l2,2.74,4.38)}\ \hspace{14cm}\]

\[\color{green}{Step\ 12:}\ \text{Determine the area of other subdivisions in the similar number and let }\ \hspace{15cm}\\ \text{A1=Area of CGJ, A3= Area of HFID and A4=Area of FEI}\ \hspace{14cm}\]

\[\color{green}{Step\ 13:}\ \text{Now calculate the area of the whole polygon}\ \hspace{17cm}\\ \text{using the command Total = A1+A2+A3+A4}\ \hspace{14cm}\]

\[\color{green}{Step\ 14:}\ \text{Verify the measurement of the area}\ \hspace{18cm}\\ \text{using the input command Area(C,J,D,I,E,F,H,G,C)} \hspace{10cm}\]