The points $\mathrm{A}(5,2), \mathrm{B}(3,4)$ and $\mathrm{C}(\mathrm{x},\mathrm{y})$ , are collinear points, and AB = BC.  Find the coordinates of C.

Given, that points A(5,2), B(3,4), and C(x, y) are collinear and AB = BC.

 B(3, 4) is the mid-point of AC.

Using the midpoint formula:

$\left(\frac{{\mathrm{x}}_1+x_2}{2},\frac{{\mathrm{y}}_1+y_2}{2}\right)=(x,y)$

$\Rightarrow \left(\frac{\mathrm{x}+5}{2},\frac{\mathrm{y}+2}{2}\right)=(3,4)$

$\Rightarrow \frac{\mathrm{x}+5}{2}=3$ and $\frac{\mathrm{y}+2}{2}=4$

$\Rightarrow \mathrm{x}+5=6$ and $\mathrm{y}+2=8$

$\Rightarrow \mathrm{x}=1$ and $\mathrm{y}=6$

Hence, the coordinates of C are (1, 6).