Question

Find the vertices of a triangle, the mid-points of whose sides are (1, 2), (3, 4), and (2, 2).

Answer 1

The triangle’s vertices are A = (0, 0), B = (2, 4), C = (4, 4)

Midpoint formula: midpoint of (x₁, y₁) and (x₂, y₂) is,

`((x_1 + x_2) / 2, (y_1 + y_2) / 2)`

If D, E, and F are the midpoints of AB, BC, and CA, respectively:

`(D = (A + B) / 2, E = (B + C) / 2, F = (C + A) / 2),`

Then let’s solve the three linear equations using the compact relations:

A = D + F − E,

B = D + E − F,

C = E + F − D.

Computation with D = (1, 2), E = (3, 4), F = (2, 2):

• A = D + F − E
  = (1 + 2 − 3, 2 + 2 − 4)
  = (0, 0)
• B = D + E − F
  = (1 + 3 − 2, 2 + 4 − 2)
  = (2, 4)
• C = E + F − D
  = (3 + 2 − 1, 4 + 2 − 2)
  = (4, 4).