(Q26) Find the area of the triangle formed by joining the mid-points of the sides of the triangle whose vertices are (0,-1),(2,1) and (0,3).Find the ratio of this area to the area of the given triangle.

ABCDEF

Given : A(0,-1) , B(2,1) and C(0,3) are the vertices of ∆ABC.

Let D,E and F be the midpoints of the sides AB BC and AC.

Midpoint (x,y) = (
x1+x2
2
 ,
y1+y2
2
 )

D = (
+
2
 ,
-+
2
 ) = (,)

E = (
+
2
 ,
+
2
 ) = (,)

F = (
+
2
 ,
-+
2
 ) = (,)

Area of a triangle ABC =

=
1
2
 |x1(y2-y3)+x2(y3-y1)+x3(y1-y2)= 0|
=
1
2
 |0( - ) + 2( + ) + 0(- - )|
=
1
2
 ||

= sq.units

Area of triangle DEF

=
1
2
 |1( - ) + 1( - ) + 0( - )|
=
1
2
 ||

= sq.units

Ratio of areas = ∆ABC:∆DEF = 4:

∆ADF ≅ ∆BED ≅ ∆DEF ≅ ∆CEF

∆ABC : ∆DEF = 4: