We know that ,
#color(blue)("If "P(x_1,y_1) ,Q(x_2,y_2),R(x_3,y_3)# are the vertices of
#color(blue)(triangle PQR#, then area of triangle:
#color(blue)(Delta=1/2||D||,# where , #color(blue)(D=|(x_1,y_1,1) ,(x_2,y_2,1),(x_3,y_3,1)|#........................#(1)#
Plot the graph as shown below.
Consider the points in order, as shown in the graph.
Let #A(2,5) ,B(5,10) ,C(10,15) and D(7,10)# be the vertices of Parallelogram #ABCD#.
We know that ,
#"Each diagonal of a parallelogram separates parallelogram"#
#"into congruent triangles."#
Let #bar(BD)# be the diagonal.
So, #triangleABD~=triangleBDC#
#:. "Area of parallelogram "ABCD=2xx "area of"triangleABD "#
Using #(1)#,we get
#color(blue)(Delta=1/2||D|| ,where, # #color(blue)(D=|(2,5,1),(5,10,1),(7,10,1)|#
Expanding we get
#:.D=2(10-10)-5(5-7)+1(50-70)#
#:.D=0+10-20=-10#
#:.Delta=1/2||-10||=||-5||#
#:.Delta=5#
#:. "Area of parallelogram "ABCD=2xx "area of"triangleABD "#
#:. "Area of parallelogram "ABCD=2xx(5)=10#
#:. "Area of parallelogram "ABCD=10 " sq. units"#
