In this picture, the mid points of the sides of the large triangle are joined to make a small triangle inside.
Calculate the coordinates of the vertices of the large triangle.
Let the three coordinates of the larger triangle be (x1, y1), (x2, y2) and (x3, y3)
The three mid points on the three sides are (3, 3) , (4, 2) , (5, 4)
According to the section formula for mid points
⇒ x1 + x2 = 6 …Equation(i)
⇒ x2 + x3 = 8 …Equation (ii)
⇒ x3 + x1 = 10 …Equation(iii)
Solving Equation (i) ,(ii) and (iii)
x2 = 6 - x1
Putting this value in equation (ii) we get
x3 - x1 = 2 …Equation (iv)
Solving Equation (iii) & (iv)
2x3 = 12
⇒ x3 = 6
Putting in Equation (iv)
6 - x1 = 2
⇒ x1 = 4
Putting this value in equation (i)
4 + x2 = 6
⇒ x2 = 2
⇒ y1 + y2 = 6 …Equation (v)
⇒ y2 + y3 = 4 …Equation (vi)
⇒ y3 + y1 = 8 …Equation (vii)
Solving Equation (v) ,(vi) and (vii)
y2 = 6 - y1
Putting this value in equation (vi) we get
y3 - y1 = - 2 …Equation (viii)
Solving Equation (vii) & (viii)
2y3 = 6
⇒ y3 = 3
Putting in Equation (viii)
3 - y1 = - 2
⇒ y1 = 5
Putting this value in equation (v)
5 + y2 = 6
⇒ y2 = 1
The Vertices of the larger triangle are (4,5) , (2,1) , (6,3)