The line joining the points $(2, x)$ and $(3, 1)$ is perpendicular to the line joining the points $(x, 4)$ and $(7, 5)$. The value of $x$ is

Slope of the line joining $(2, x)$ and $(3, 1)$ is $\frac{1-x}{3-2}$ and that of the line joining $\left(x,4\right)$ and $\left(7,5\right)$ is $\frac{5-4}{7-x}.$ Since the lines are perpendicular the product of the slopes is $-1$

$\begin{array}{l}\Rightarrow \frac{1-x}{1}\times \frac{1}{7-x}=-1\\ \Rightarrow x=4.\end{array}$