Question

If the coordinates of the mid-points of the sides of a triangle are (3, 4), (4, 6) and (5, 7), find its vertices.

Answer 1

Step-by-step explanation:

Let the vertices of the triangle be A(x₁ , y₁ ), B(x₂ ,y₂ ) and C(x₃ ,y₃ ).

Let the mid-points of the side BC be D (3, 4)

=> (x₂ + x₃/2, y₃ + y₂/2) = (3, 4)

=> x₂ + x₃ = 6,y₃ + y₂ = 8-------(1)

Let the mid-points of the side AC be E (4, 6)

=>(x₁ + x₃/2, y₁ + y₃/2) = (4, 6)

=>x₁ + x₃ = 8, y₁ + y₃ = 12----(2)

Let the mid-points of the side AB be F (5, 7).

=> (x₁ + x₂/2, y₁ + y₂/2) = (5, 7)

=>x₁ + x₂ = 10, y₁ + y₂ = 14----(3)

We know that centroid of the triangle formed by the midpoints of the sides of the triangle will be same as the centroid of that triangle.

Hence, we get

G(x₁ + x₂ + x₃/3, y₁ + y₂ + y₃/3) = (4, 17/3)

=>x₁ + x₂ + x₃ = 12

y₁ + y₂ + y₃ = 17---(*)

Now,

Using (*) and (1), we find that A(x₁, y₁) = (6, 9)

Using (*) and (2), we find that B(x₂, y₂) = (4, 5)

Using (*) and (3), we find that C(x₃, y₃) = (2, 3).

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