Question

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

Answer 1

Question

Find the area of the triangle formed by joining the mid-point 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.

ANSWER

since D,E and F are the midpoints of AB, BC and CA of triangle respectively

coordinate of D= (1,0)

$(\frac{0 + 2}{2} \frac{ - 1 + 1}{2})$

Coordinate of E= (1,2)

$( \frac{2 + 0}{2} \: \frac{1 + 3}{2})$

Coordinate of F= (0,1)

$( \frac{0 + 0}{2} \: \frac{ - 1 + 3}{2} )$

∴ar(△DEF)= 1/2[1(2−1)+1(1−0)+0(0−2)]=1sq.unit

and

ar(△ABC)= 1/2[0(1−3)+2(3+1)+0(−1−1)]=4sq.unit

$\frac{area \: def}{area \: abc} = \frac{1}{4}$

⇒ar(△DEF):ar(△ABC)=1:4