If the vertices of a triangle have integer co | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(B) Let the vertices of the triangle be $(x_1, y_1), (x_2, y_2),$ and $(x_3, y_3)$ where all coordinates are integers.The area of a triangle with inte

Vedclass · Accepted Answer

Equilateral

Vedclass · Answer

(B) Let the vertices of the triangle be $(x_1, y_1), (x_2, y_2),$ and $(x_3, y_3)$ where all coordinates are integers.The area of a triangle with integer coordinates is given by the formula $	ext{Area} = \frac{1}{2} |x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2)|$,which is always a rational number (specifically,a multiple of $0.5$).If the triangle were equilateral with side length $a$,its area would be $\frac{\sqrt{3}}{4} a^2$.Since the coordinates are integers,the square of the side length $a^2 = (x_1 - x_2)^2 + (y_1 - y_2)^2$ must be an integer.Thus,the area $\frac{\sqrt{3}}{4} a^2$ would be an irrational number because $\sqrt{3}$ is irrational and $a^2$ is an integer.Since a triangle cannot have an area that is both rational and irrational,a triangle with integer coordinates can never be equilateral.

If the vertices of a triangle have integer coordinates,then what type of triangle can it never be?

Similar Questions

The points $(0, 0)$,$(a, 0)$,and $\left( \frac{a}{2}, \frac{a\sqrt{3}}{2} \right)$ are vertices of:

The bisectors of the angles of a parallelogram enclose a

If $\Delta_1$ is the area of the triangle formed by the centroid and two vertices of a triangle,and $\Delta_2$ is the area of the triangle formed by the mid-points of the sides of the same triangle,then $\Delta_1 : \Delta_2 =$

$A$ rectangle $ABCD$ has its side parallel to the line $y=2x$ and vertices $A, B, D$ are on lines $y=1, x=1$ and $x=-1$ respectively. The coordinate of $C$ can be

The area of a parallelogram formed by the lines $ax \pm by \pm c = 0$ is

Vedclass Test Series

Exam Paper Generator

Online Exam Module