Final answer:
Option A, p → q, is the original conditional statement stating that doubling the dimensions of a rectangle leads to an area increase by a factor of four. Option C, q → p, is the original statement's converse, suggesting that if the area increases by four, the dimensions doubled. These two are the correct options that correspond to the relationship between a rectangle's dimensions and area. Thus, the correct options that represent the mathematical property are A and C.
Step-by-step explanation:
When we talk about the mathematical properties of a rectangle's dimensions and the corresponding area, we engage with the concept known as scale factors. Doubling the dimensions of a rectangle (represented as p) indeed causes the area to increase by a factor of four (represented as q). Understanding this relationship allows us to analyze the statements provided in the question.
Firstly, p → q represents the original conditional statement. If the rectangle's dimensions are doubled, then the area increases by a factor of four. Secondly, the converse of the original conditional statement would be q → p, which essentially flips the implication of the original statement— if the area increases by a factor of four, then the rectangle's dimensions are doubled.
Now let's clarify the terms inverse and contrapositive. The inverse of a conditional statement negates both the hypothesis and the conclusion. In this case, the inverse would be represented as ~p → ~q. The contrapositive, on the other hand, both negates and reverses the hypothesis and conclusion of the original statement. However, none of the options provided denote the contrapositive of the original conditional statement. Thus, the correct options that represent the mathematical property are A and C.