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Question

If two vertices of an equilateral triangle have integral coordinates, then the third vertex will have:
  1. integral coordinates
  2. coordinates which are rational
  3. at least one coordinate irrational
  4. coordinates which are irrational

A
integral coordinates
B
coordinates which are rational
C
coordinates which are irrational
D
at least one coordinate irrational
Solution
Verified by Toppr

Let the vertices of the equilateral triangle be (x1,y1),(x2,y2) and (x3,y3)
If none of xi and yi(i=1,2,3) are irrational, then
area of Δ=12∣ ∣ ∣x1y11x2y21x3y31∣ ∣ ∣= rational
But the area of an equilateral triangle =34(side)2= irrational
Thus, the two statements are contradictory,
Therefore, both the coordinates of the third vertex cannot be rational.

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