Question

find the area of triangle formed by joining the midpoint of the sides of the triangle whose vertices are zero, - 1, 2, 1 and 0, 3 find the ratio of this area to the area of the given Triangle

Answer 1

Let the vertices of the triangle be A (0, −1), B (2, 1), C (0, 3).

Let D, E, F be the mid-points of the sides of this triangle. Coordinates of D, E, and F are given by

D=[(0+2)/2 , (-1+1) /2] =[1,0]

E=  [(0+0)/2 , (3-1) /2] =[0,1]

F=[(2+0)/2 , (1+3) /2] =[1,2  ]

Area of triangle =1/2{x1[y2-y3] +x2[yy1]+x3[y1-y2]}                                                    

Area of triangle DEF =1/2{1[2-1] +1[1-0]+0[0-2]}

Area of triangle DEF =1/2{1+1} =1 square units..............[1]

Area of triangle ABC =1/2{0[1-3] +2(3-[-1]+0[-1-1]}

Area of triangle ABC =1/2{8} =4square units....................[2]

from [1] and [2] we get the ratio

∴ the requared ratio =1:4 ANS