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Question

For △ABC whose vertices are given by A(3,6),B(−4,7) and C(10,−7), if A1,B1,C1 represent the mid points of sides opposite to vertices A,B,C respectively, of △ABC and A2,B2,C2 represent the mid points of sides opposite to vertices A1,B1,C1 respectively, of △A1B1C1 and so on, then which of the following is/are correct?

A
The centroid of A2B2C2 is (3,2)
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B
The centroid of A3B3C3 is (3,2)
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C
Area (A2B2C2)Area (ABC)=116
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D
Area (A3B3C3)Area (ABC)=116
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Solution

The correct option is C Area (A2B2C2)Area (ABC)=116
Given vertices of ABC are A(3,6),B(4,7) and C(10,7)

We know that,
Centroid of triangle = Centroid of triangle formed by midpoints.
So, centroid of triangles ABC=A1B1C1=A2B2C2=(34+103,6+773)=(3,2)


Area of triangle formed by midpoints =14×Area of original triangle.

So, Area (A2B2C2)Area (A1B1C1)=14
and Area (A1B1C1)Area (ABC)=14
Area (A2B2C2)Area (ABC)=116

Similarly, Area (A3B3C3)Area (ABC)=164

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