Question

1. in a set of similar objects, why does any one object have unique scale factors relating it to every other object in the set?

Explanation:

Similar objects have proportional corresponding dimensions. The scale factor between two similar objects is the ratio of their corresponding linear measurements. Since each object in the set has a distinct size (even though they are similar), when comparing one object to another, the ratio of their corresponding lengths (the scale factor) will be unique for each pair of objects. For example, if Object A has a length of \(l_A\), Object B has \(l_B\), and Object C has \(l_C\) (\(l_A
eq l_B
eq l_C\)), the scale factor from A to B is \(\frac{l_B}{l_A}\) and from A to C is \(\frac{l_C}{l_A}\), which are different values if \(l_B
eq l_C\).

Answer:

Because the ratio of corresponding linear dimensions (the scale factor) between one object and each of the other objects in the set is unique due to each object having distinct (but proportional) sizes.