Each of the related words indicates a certain kind of physical “roominess.” That is, they pertain to some three-dimensional (or perhaps two-dimensional) size or measure that is large and open. The problem with the word “lengthened” is that it really only deals with an extension in a single dimension. Even if you lengthen a board of wood, you are referring to the act of making it longer in one direction. Because of this word “lengthened” does not indicate “roominess” as much as it does merely the act of making something longer.

All of the related words pertain to health in some way, even if they indicate lack of health. To be vigorous is to be strong and healthy, while to be frail and ailing is to be weak and / or sick. Only “tepid” is not related to this group of words, as it means “lukewarm” or “lacking excitement or enthusiasm.” While this describes a certain state of mind for a person, it is not directly related to health—not even mental health, for someone could be tepid merely because something really is not interesting.

All of the related words somehow are related to the lungs. In order to see this, it is necessary to know the meanings of pulmonary and alveolar. The alveoli are the tiny “sacks” in the lung that allow for blood and air to interact. The word “pulmonary” means “related to the lungs.” (It is derived from the Latin for “lungs”). These words are not all synonymous with the word “lung,” but they are all related thereto. In contrast, “cardiac” describes something related to the heart.

Each of the related words pertain to illegality in some way. A miscreant is a person who breaks the law—though the term is sometimes used also to describe someone who acts badly in general. A delinquent is a person who commits crimes, usually small ones. Often, you will hear the word combined with “juvenile” as in “juvenile delinquent,” which is used to describe young people who commit minor crimes (like vandalizing with paint or eggs). The word “unaccounted” means “not taken to mind” in some consideration (calculation, etc). This is not related to the other words.

All of the related words describe financial matters in a general fashion. The case of “monetary” is a bit more specific, as it deals in particular with things pertaining to the currency of a nation. (The monetary policy of the United States is the policy regarding the value of the US dollar). Pecuniary means “pertaining financial matters.” It comes from the Latin word for cattle, which would have been a form of wealth. The word “retail” at best is related to a type of sales location. Retail sales are made for use (not resale, as in things sold to stores in order to sell them). This is not strictly financial but signifies a type of location at which financial transactions occur.

All of the related words in this question are related to being sad in some way. This is obvious in the case of depressed and gloomy. It is at least a significant possibility in the case of mourning. (Perhaps someone could mourn without being sad, but that seems rare if not impossible). Someone who is doleful is extremely sad. Thus only consoled stands out in this group, as it means “having been comforted.”

The set of related words in this question all are somehow related to the word “aggressive” or at least “fighting.”  Someone who is “belligerent” is aggressive. Belligerence is the act of being very nasty and aggressive, as when someone verbally abuses another person. The word literally means “warlike,” from the Latin for “war.” The expression “antebellum South” literally means “the southern states (in the US) before (ante-) the Civil War (-bellum).” To be pugnacious is to be quarrelsome. It is related to the word “pugilist,” which means “boxer” (as in the type of fist-fighter). Only “dispassionate” describes someone who is not aggressive. A dispassionate person has little or no passions; that is, they are emotionless.

The general notion uniting the related words here is that of not being practical or feasible. This might be hard to see in the case of “academic,” which can of course mean “pertaining to education;” however, one can also say, “That is really an academic question and is not important for what we need to do here and now.” Whether or not such a statement is true, it implies that something is not currently relevant for a practical undertaking. This is another possible meaning for “academic”—as though to say, “That should be handled in the so-called ivory tower of academia.” Only “dedicated” is unrelated to the others, for it does not indicate anything about being practical but merely being devoted. (Someone can be quite dedicated to a purely “academic” affair).

All of the related words pertain to states that a fluid could have. Even something “gelatinous” is like a slow flowing fluid. Something viscous is thick but still fluid. Certainly, thin and runny substances are fluids as well. Often we call them “watery” as if to say, “They run like water.” Something solid, on the other hand, is not a fluid, whatever else might be said. While something “gelatinous” might be thought of as being solid, there is still a qualitative difference between pure solidity and being gelatinous. The latter could at least conceivably be applied to a fluid.

Note first that all of these words have something to do with being cold or frozen. The one word that is likely unknown is “gelid,” which means “extremely cold.” Do not confuse this with “gel.” Interestingly, the two words are related to the same Latin root for “frost” or “extreme cold,” but “gelid” only means “extremely cold.” Now, all of the related words describe states of extreme cold or at least the state of being very cold. In contrast with all of those, the word “cold” is less intense, thus fitting least with the rest of the words.