Question

the midpoints D, E and F of the sides AB, BC and CA of a triangle are (3,4), (8,9) and (6,7) respectively. find the coordinates of the vertices of the triangle.

Answer 1

D(2, 1), E(1, 0), F(1, 3)

let the vertices be A(x1, y1), B(x2, y2), C(x3, y3).

By applying midpoint formula:

(x1 + x2)/2 = 2

 x1 + x2 = 4 ...(1)

(x2 + x3)/2 = 1

 x2 + x3 = 2 ...(2)

(x1 + x3)/2 = 1

 x1+ x3= 2 ...(3)

 x1 = 2  x3

(1)  2  x3+ x2= 4

 x2 x3= 6 ...(4)

On adding (2) and (4), we obtain

2x2= 8

 x2= 4

∴ 4+x3= 2

 x3= 2

∴ x1+ 4 = 4

 x1= 0

Similarly for y 

(y1 + y2)/2 = 1

 y1 + y2 = 2 ...(5)

(y2 + y3)/2 = 0

 y2 + y3 = 0 ...(6)

(y1 + y3)/2 = 3

 y1 + y3 = 6 ...(7)

(6) y2 = y3 

(5) y1 y3= 2 ...(8)

On adding (7) and (8), we obtain

2y1= 8

 y1= 4

∴ 4  y3= 2

 y3= 2

∴ y2= 2

The vertex A, B and C are (0, 4), (4, 2) and (2, 2)

Answer 2

Plz see the attachment below