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Mid Point

A-4, 2, B0, 2...

Question

A(-4, 2), B(0, 2) and C(-2, -4) are vertices of a triangle ABC. P, Q and R are mid-points of sides BC, CA and AB respectively. Show that the centroid of $Δ$ PQR is the same as the centroid of $Δ$ ABC.

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Solution

$C e n t r o i d space o f space A B C fraction numerator x subscript 1 plus x subscript 2 plus x subscript 3 over denominator 3 end fraction space equals space fraction numerator negative 4 plus 0 plus left parenthesis negative 2 right parenthesis over denominator 3 end fraction space equals space minus 2 a n d fraction numerator y subscript 1 plus y subscript 2 plus y subscript 3 over denominator 3 end fraction space equals space fraction numerator 2 plus 2 minus 4 over denominator 3 end fraction space equals space 0$
$m p space o f space A B space equals space open parentheses fraction numerator negative 4 plus 0 over denominator 2 end fraction comma space fraction numerator 2 plus 2 over denominator 2 end fraction close parentheses space equals space open parentheses negative 2 comma space 2 close parentheses m p space o f space B C space equals space open parentheses fraction numerator 0 plus negative 2 over denominator 2 end fraction comma space fraction numerator 2 plus left parenthesis negative 4 right parenthesis over denominator 2 end fraction close parentheses space equals space open parentheses negative 1 comma space minus 1 close parentheses m p space o f space C A space equals space space open parentheses fraction numerator negative 2 plus negative 4 over denominator 2 end fraction comma space fraction numerator negative 4 plus 2 over denominator 2 end fraction close parentheses space equals space open parentheses negative 3 comma space minus 1 close parentheses c e n t r o i d space o f space P Q R space equals space fraction numerator negative 2 minus 1 minus 3 over denominator 3 end fraction comma space fraction numerator 2 minus 1 minus 1 over denominator 3 end fraction space equals space open parentheses negative 2 comma space 0 close parentheses$

hence the centroid of ABC and PQR are same

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