Is the sentence
a correct translation of
One of my family members is thin and cute
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In this case, "私の家族の一人は" is correct. Although it might feel more natural to specify which one(brother, sister, father, etc...).
However, the following change should be made :
痩せて > 痩せていて
Adding the "いて" is needed to reflect the ongoing state of being thin rather than the plan or the action of loosing weight.