Question

If the each of the vertices of a triangle has integral coordinates, 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

Option A:

If we take $(0,0)$ as one of the vertices, then by taking integral coordinates on x-axis and y-axis, we can form right-angled triangles. For example $(0,0),(3,0),(0,5)$ will form a right angled triangle.

Hence, option A is possible,

Option B:

Let $(x_1,y_1),(x_2,y_2),(x_3,y_3)$ are the coordinates of the three vertices.

Then area is given by $\left|\begin{array}{ccc}1& 1& 1\\ x_1& x_2& x_3\\ y_1& y_2& y_3\end{array}\right|$.

If all the coordinates are integers then the value of the above determinant is a rational number.....(1)

Now, if the above triangle is an equilateral triangle, then area can also be expressed as $\frac{\sqrt{3}}{4}\times sid{e}^2$

For integer coordinates $sid{e}^2$ will be a positive integer. Hence $\frac{\sqrt{3}}{4}\times sid{e}^2$ is an irrational number....(2).

Since (1) and (2) contradict each other, the triangle can not be equilateral.

Option C:

Again we can take $(0,0)$ and $(4,0)$ as two points and any third point on $x=2$ will form an isosceles triangle along with $(0,0)$ and $(4,0)$. For example: $(0,0)$, $(4,0)$ and $(2,6)$ will form an isosceles triangle.

Hence, option C is possible.