Shortest side

Since the ratio is $\dfrac{1}{2}:\dfrac{1}{3}:\dfrac{1}{4}$ , the total is $\dfrac{1}{2}+\dfrac{1}{3}+\dfrac{1}{4}=\dfrac{13}{12}$ .

Now we know that $\dfrac{1}{4}$ is the smallest, so the length of the shortest side is

$\dfrac{\left(\dfrac{1}{4}\right)}{\left(\dfrac{13}{12}\right)} \times 52=\dfrac{1}{4} \times \dfrac{12}{13} \times 52=$ $\color{#D61F06}\large \boxed{12}$

Let the side lengths be $a$ , $b$ and $c$ , then $a:b:c = \dfrac 12 : \dfrac 13 : \dfrac 14 = \dfrac {12}2 : \dfrac {12}3 : \dfrac {12}4 = 6: 4: 3$ . The shortest length $c = \dfrac c{a+b+c} \times (a+b+c) = \dfrac 3{6+4+3} \times 52 = \boxed{12}$ .

Since, the sides of a triangle are in the ratio

$\dfrac{1}{2}$ $:$ $\dfrac{1}{3}$ $:$ $\dfrac{1}{4}$

Multiply the three by $12$ ,

$\dfrac{1}{2}$ $\times 12$ $:$ $\dfrac{1}{3}$ $\times 12$ $:$ $\dfrac{1}{4}$ $\times 12 = 6:4:3$

Suppose, the sides are $6x$ $cm,$ $4x$ $cm$ and $3x$ $cm.$

The perimeter is $52$ $cm$

So, $6x + 4x + 3x = 52$

$⇒ 13x = 52$

$⇒ x = 4$

Therefore the length of the shortest side is $3x$ $cm$ $= 12$ $cm$