question_answer

In Fig. 6, ABC is a triangle coordinates of whose vertex A are (\[0,\text{ }1\]). D and E respectively are the mid-points of the sides AB and AC and their coordinates are (1, 0) and (0, 1) respectively. If F is the mid-point of BC, find the areas of \[\Delta \text{ }ABC\] and \[\Delta \text{ }DEF\] .

Answer:

Given, the coordinates of vertex \[A\text{ (}0,-1)\] and, mid points \[D\text{ (}1,0)\] and \[E(0,1)\] respectively. Since, D is the mid-point of AB

Let, coordinates of B are (x, y)

then, \[\left( \frac{x+0}{2},\frac{y-1}{2} \right)=(1,0)\]

which gives \[B\,(2,1)\]

Similarly, E is the mid-point of AC

Let, coordinates of C are \[(x,y)\]

then, \[\left( \frac{x'+0}{2},\frac{y'-1}{2} \right)=(0,1)\]

which gives C (0, 3)

Now, Area of \[\Delta \,ABC=\frac{1}{2}|[0(1-3)+2(3+1)+0(-1-1)]|\]

\[=4\] sq. units.

Now, F is the mid-point of BC.

\[\Rightarrow \] Coordinates of F are \[\left( \frac{2+0}{2},\frac{1+3}{2} \right)=(1,2)\]

\[\therefore \]      Area of \[\Delta \,DEF=\frac{1}{2}|[1(1-2)+0(2-0)+1(0-1)]|\]

\[=\frac{|-2|}{2}=1\,\]sq. unit