The co-ordinates of the vertices of a triangle are [m(m + 1), (m + 1)], [(m + 1)(m + 2), (m + 2)] and [(m + 2)(m + 3), (m + 3)]. Then which one among the following is correct?

The correct option is: The area of the triangle is independent on m.

To understand why this option is correct, let us calculate the area of the triangle using the given coordinates of its vertices.

Let the coordinates of the vertices of the triangle be:

$$A=\left(m\left(m+1\right),m+1\right)$$

$$B=\left(\left(m+1\right)\left(m+2\right),m+2\right)$$

$$C=\left(\left(m+2\right)\left(m+3\right),m+3\right)$$

We can identify these coordinates as:
$\left(x_1,y_1\right)=\left(m^2+m,m+1\right)$
$\left(x_2,y_2\right)=\left(m^2+3m+2,m+2\right)$
$\left(x_3,y_3\right)=\left(m^2+5m+6,m+3\right)$

The standard formula for the area of a triangle with vertices
$\left(x_1,y_1\right)$
,
$\left(x_2,{\mathrm{y<!--="2"</msub-->}}_{}\right)$
, and
$\left(x_3,{\mathrm{y<!--="3"</msub-->}}_{}\right)$
is given by:

$$\text{Area}=\frac{1}{2}\&InvisibleTimes;|x_1\left(y_2-y_3\right)+x_2\left(y_3-y_1\right)+x_3\left(y_1-y_2\right)|$$

Let us calculate the differences in the y-coordinates first:

$$y_2-y_3=\left(m+2\right)-\left(m+3\right)=-1$$

$$y_3-y_1=\left(m+3\right)-\left(m+1\right)=2$$

$$y_1-y_2=\left(m+1\right)-\left(m+2\right)=-1$$

Now, let us substitute these values and the x-coordinates back into the area formula:

$$\text{Area}=\frac{1}{2}\&InvisibleTimes;|\left(m^2+m\right)\left(-1\right)+\left(m^2+3m+2\right)\left(2\right)+\left(m^2+5m+6\right)\left(-1\right)|$$

Let us expand the terms inside the absolute value brackets:

$$-\left(m^2+m\right)+2\left(m^2+3m+2\right)-\left(m^2+5m+6\right)$$

$$=-m^2-m+2m^2+6m+4-m^2-5m-6$$

Grouping the like terms together:

$$\text{For}{m}^2\text{terms:}-m^2+2m^2-m^2=0$$

$$\text{For}m\text{terms:}-m+6m-5m=0$$

$$\text{For constant terms:}4-6=-2$$

Substituting these back into the expression, the term inside the absolute value simplifies to:

$$0+0-2=-2$$

Now, we calculate the area:

$$\text{Area}=\frac{1}{2}\&InvisibleTimes;|-2|$$

$$\text{Area}=\frac{1}{2}\&InvisibleTimes;2=1$$

Since the area of the triangle is a constant value of 1 square unit, it does not depend on the value of
$m$
at all. Therefore, the area of the triangle is independent of
$m$
.