Question 3 Find area of a triangle formed by joining mid-points of sides of another triangle whose vertices are (0, -1), (2, 1) and (0, 3). Find ratio of this area to the area of the given triangle.
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Solution
Vertices of the given triangle are A (0,−1),B(2,1),C(0,3). Let D, E, F be mid-points of its sides. Coordinates of D, E, and F are given by D=(0+22,−1+12)=(1,0) E=(0+02,3−12)=(0,1) F=(2+02,1+32)=(1,2) Area of a triangle = 12{x1(y2−y3)+x2(y3−y1)+x3(y1−y2)} Area of ΔDEF=12{1(2−1)+1(1−0)+0(0−2)} =12(1+1)=1squareunits Area of ΔABC=12[0(1−3)+2{3−(−1)}+0(−1−1)] =12{8}=4squareunits Therefore, the required ratio is 1:4.