Answer :

Let the three coordinates of the larger triangle be (x_{1}, y_{1}), (x_{2}, y_{2}) and (x_{3}, y_{3})

The three mid points on the three sides are (3, 3) , (4, 2) , (5, 4)

According to the section formula for mid points

⇒ x_{1} + x_{2} = 6 …Equation(i)

⇒ x_{2} + x_{3} = 8 …Equation (ii)

⇒ x_{3} + x_{1} = 10 …Equation(iii)

Solving Equation (i) ,(ii) and (iii)

x_{2} = 6 - x_{1}

Putting this value in equation (ii) we get

x_{3} - x_{1} = 2 …Equation (iv)

Solving Equation (iii) & (iv)

2x_{3} = 12

⇒ x_{3} = 6

Putting in Equation (iv)

6 - x_{1} = 2

⇒ x_{1} = 4

Putting this value in equation (i)

4 + x_{2} = 6

⇒ x_{2} = 2

⇒ y_{1} + y_{2} = 6 …Equation (v)

⇒ y_{2} + y_{3} = 4 …Equation (vi)

⇒ y_{3} + y_{1} = 8 …Equation (vii)

Solving Equation (v) ,(vi) and (vii)

y_{2} = 6 - y_{1}

Putting this value in equation (vi) we get

y_{3} - y_{1} = - 2 …Equation (viii)

Solving Equation (vii) & (viii)

2y_{3} = 6

⇒ y_{3} = 3

Putting in Equation (viii)

3 - y_{1} = - 2

⇒ y_{1} = 5

Putting this value in equation (v)

5 + y_{2} = 6

⇒ y_{2} = 1

The Vertices of the larger triangle are (4,5) , (2,1) , (6,3)

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