Answer :
given , AL is height of TriangleABD , CM is the height of triangleDBC , BD is the common base of both the triangleABD & TriangleDBC .
to prove - 1/2*BD* AL + CM
proof - in triangle ABD ,
area = [tex] \frac{1}{2} * base * height [/tex]
[tex] \frac{1}{2} * BD * AL[/tex] ---------(eqn .i)
in triangle DBC,
area = [tex] \frac{1}{2} *base*height [/tex]
= [tex] \frac{1}{2} * DB * CM[/tex] ------------- ( eqn. ii )
on adding eqn. i & ii , we get ,
area of quad. ABCD = ar.( traingle ABD + triangle DBC )
= [tex] \frac{1}{2}+\frac{1}{2} * BD + BD * AL + CM [/tex]
= [tex] \frac{1}{2}+\frac{1}{2} * 2BD * AL + CM [/tex]
= [tex] \frac{1}{2} * BD ( AL + CM ) [/tex]
to prove - 1/2*BD* AL + CM
proof - in triangle ABD ,
area = [tex] \frac{1}{2} * base * height [/tex]
[tex] \frac{1}{2} * BD * AL[/tex] ---------(eqn .i)
in triangle DBC,
area = [tex] \frac{1}{2} *base*height [/tex]
= [tex] \frac{1}{2} * DB * CM[/tex] ------------- ( eqn. ii )
on adding eqn. i & ii , we get ,
area of quad. ABCD = ar.( traingle ABD + triangle DBC )
= [tex] \frac{1}{2}+\frac{1}{2} * BD + BD * AL + CM [/tex]
= [tex] \frac{1}{2}+\frac{1}{2} * 2BD * AL + CM [/tex]
= [tex] \frac{1}{2} * BD ( AL + CM ) [/tex]
The area of figure ABCD = ar(ΔABD)+ar(ΔCBD)
=⇒⇒⇒[tex] \frac{1}{2} X BD X AL + \frac{1}{2} X BD X CM \\ \frac{1}{2} X BD (AL+CM) \\ Area of the Quad. ABCD=\frac{1}{2} X BD (AL+CM) [/tex]
=⇒⇒⇒[tex] \frac{1}{2} X BD X AL + \frac{1}{2} X BD X CM \\ \frac{1}{2} X BD (AL+CM) \\ Area of the Quad. ABCD=\frac{1}{2} X BD (AL+CM) [/tex]