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Question: If in triangle ABC, A = (1, 10), circumcentre = \(\left( - \frac{1}{3},\frac{2}{3} \right)\)and orth...

If in triangle ABC, A = (1, 10), circumcentre = (−13,23)\left( - \frac{1}{3},\frac{2}{3} \right)and orthocentre = (113,43)\left( \frac{11}{3},\frac{4}{3} \right), then the coordinates of mid-point of side opposite to A is

A

(1, – 11/3)

B

(1, 5)

C

(1, – 3)

D

(1, 6)

Answer

(1, – 11/3)

Explanation

Solution

Circumcentre O ≡ (−13,23)\left( - \frac{1}{3},\frac{2}{3} \right)

and orthocenter H ŗ (113,43)\left( \frac{11}{3},\frac{4}{3} \right)

\coordinates of G are(1,89)\left( 1,\frac{8}{9} \right)

A(1, 10), G(1,89)\left( 1,\frac{8}{9} \right)

AD : DG = 3 : –1

Dx = 3−12\frac{3 - 1}{2}= 1

Dy = 83−102=−113\frac{\frac{8}{3} - 10}{2} = - \frac{11}{3}

\coordinate of the mid point is BC are (1,−113)\left( 1, - \frac{11}{3} \right)