Final answer:
To find the length x of JK in similar pentagons, we need to use the properties of similarity and proportions. The ratio of corresponding sides would give us x, provided we have the side lengths or the similarity ratio between the pentagons.
Step-by-step explanation:
To find the length of x of JK, one must consider the properties of similar pentagons ABCDE and JKLMN. Similar figures have the same shape but not necessarily the same size and corresponding sides are in proportion. According to the provided information, we have AB = 3x when AC = 3R. Also, in similar figures angles are equal, so if we have additional angle measures, we can confirm that similarity and find the lengths using proportions.
It is mentioned that triangles HKD and KFD are congruent, which suggests that triangle AHD is also congruent and has the side AC = 3R. If AB = 3x, and these pentagons are similar, the ratio of corresponding sides AB (of ABCDE) to JK (of JKLMN) will give us the length of x, as the ratio has to be constant due to the similarity of the figures.
Without the first pentagon's explicit dimensions or the similarity ratio between the two pentagons, we can't compute a numerical answer for x. However, we know that if the information were provided, one would set up a proportion such as AB/AC = JK/JL and solve for the unknown length x.
As for the other scenarios described, such as estimating distances using proportionality with human body parts or understanding the areas of squares, those also rely on the concept of proportional reasoning and can be helpful in various real-world applications to estimate measurements when direct measurement is not feasible.