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Question
If ᐃ1 is the area of the triangle formed by the centroid and two vertices of a triangle, ᐃ2 is the area of the triangle formed by the mid-points of the sides of the same triangle, then ᐃ1 : ᐃ2 =
Options
A.3 : 4
B.4 : 1
C.4 : 3
D.2 : 1
Solution
Let A(x1, y1), B(x2, y2) and C(x3, y3) be the vertices of a ᐃABC, and let G be its centroid.
Then,
ᐃ1 = Area of ᐃGBC
⇒ ᐃ1 =
, where ᐃ is the area of ᐃABC
ᐃ2 = Area of triangle formed by the mid-points of the sides
⇒ ᐃ2 =
ᐃ
∴ ᐃ1 : ᐃ2 = 4 : 3
Then,
ᐃ1 = Area of ᐃGBC
⇒ ᐃ1 =
, where ᐃ is the area of ᐃABCᐃ2 = Area of triangle formed by the mid-points of the sides
⇒ ᐃ2 =
ᐃ∴ ᐃ1 : ᐃ2 = 4 : 3
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