The sides of a right-angled triangle are in arithmetic progression....

Let three sides of triangle be: $a - d, a, a + d$.

Where, $d > 0, a > 0$.

$⇒$ length of smallest side $= a - d$ units

Now, ${( a + d )}^{2} = {a}^{2} + {( a - d )}^{2}$

$⇒ a\left(a - 4d\right) = 0$

$∴ a = 4d$ .....(i)

(As $a = 0$ is rejected)

Also, $\dfrac{1}{2}a . \left(a - d\right) = 24$

$⇒ a\left(a - d\right) = 48$ .....(ii)

$∴$ From (i) and (ii), we get  $a = 8, d = 2$

Hence, length of smallest side is

\left(a - d\right) = \left(8 - 2\right) = 6$units