Question

If the vertices of a triangle have integral coordinates, then the triangle can not be equilateral.

विकल्प

• True

• False

Answer 1

This statement is True.

Explanation:

Let A(x₁, y₁), B(x₂, y₂) and C(x₃, y₃) be the vertices of a triangle ABC, where x_i, y_i, i = 1, 2, 3 are integers.

Then, the area of ΔABC is given by

Δ = `1/2 [x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2)]`

= A rational number   ......[∵ x_i, y_i, are integers]

If possible, let the triangle ANC be an equilateral triangle

Then its area is given by

Δ = `sqrt(3)/4 ("side")^2 = sqrt(3)/4 (AB)^2`  ......[∵ AB = BC = CA]

= `sqrt(3)/4 (a  "positive integer")`   .....[∵ verticles are integral∴ AB² is a integer]

= an irrational number

This is a contradiction to the fact that the area is a rational number.

Hence, the triangle cannot be equilateral.