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

The correct option is D The centroid of A2B2C2 is (3,2)
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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