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(D) The ratio of the sides is given as $a: b: c = \frac{1}{2}: \frac{1}{3}: \frac{1}{4}$.To simplify the ratio,multiply each term by the least common

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$12$

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(D) The ratio of the sides is given as $a: b: c = \frac{1}{2}: \frac{1}{3}: \frac{1}{4}$.To simplify the ratio,multiply each term by the least common multiple $(LCM)$ of the denominators $(2, 3, 4)$,which is $12$.$a: b: c = (\frac{1}{2} 	imes 12) : (\frac{1}{3} 	imes 12) : (\frac{1}{4} 	imes 12) = 6: 4: 3$.Let the sides be $6x, 4x,$ and $3x$ centimeters.The perimeter of the triangle is the sum of its sides,which is given as $52 	ext{ cm}$.$6x + 4x + 3x = 52$.$13x = 52$.$x = \frac{52}{13} = 4$.The smallest side corresponds to the smallest ratio value,which is $3x$.Smallest side $= 3 	imes 4 = 12 	ext{ cm}$.

The sides of a triangle are in the ratio of $\frac{1}{2}: \frac{1}{3}: \frac{1}{4}$. If the perimeter is $52 \text{ cm}$,then the length of the smallest side is ........ $\text{cm}$.

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