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Question
If the vertices of a triangle have integral coordinates, then the triangle can not be equilateral.
Options
True
False
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Solution
This statement is True.
Explanation:
Let A(x1, y1), B(x2, y2) and C(x3, y3) be the vertices of a triangle ABC, where xi, yi, 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 ......[∵ xi, yi, 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.
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