Question

If each of the vertices of a triangle has integral co-ordinates then the triangle may be

• A. Right angled
• B. Equilateral
• C. Isosceles
• D. None of these

Answer 1

Answer: (A), (C)

The correct options are
A Right angled
C Isosceles

In an equilateral triangle, at least one of the vertices has irrational co-ordinate.
Hence a triangle with all rational and integral co-ordinates cannot be an equilateral triangle.
Now considering an isosceles triangle,
Let one vertice be $(0,0)$ and another vertice be $(a,0)$.
Then the locus of the third vertice is the perpendicular bisector of the line joining $(0,0)$ and $(a,0)$.
Hence it can have the coordinates of the third vertice as integral coordinates.
Now coming to right angled triangle, consider the set of points $(0,0),(3,0),(0,4)$.
Hence it too can have integral coefficients.