## Answer :

**Original Conditional Statement [tex]\( p \rightarrow q \)[/tex]:**

- "If we double the dimensions of a rectangle, then the area increases by a factor of 4."

- This statement is given to be true.

**Inverse [tex]\( \sim p \rightarrow \sim q \)[/tex]:**

- "If we do not double the dimensions of a rectangle, then the area does not increase by a factor of 4."

- This is not necessarily true. The rectangle could change dimensions in some other way that still results in the area increasing by a factor of 4.

**Converse [tex]\( q \rightarrow p \)[/tex]:**

- "If the area increases by a factor of 4, then we have doubled the dimensions of the rectangle."

- This assumes that increasing the area by a factor of 4 can only be achieved by doubling the dimensions. There might be other ways to achieve this area increase, so this is not necessarily true.

**Contrapositive [tex]\( \sim q \rightarrow \sim p \)[/tex]:**

- "If the area does not increase by a factor of 4, then we have not doubled the dimensions of the rectangle."

- This is logically equivalent to the original conditional statement [tex]\( p \rightarrow q \)[/tex], and thus true.

Based on these logical relationships and the provided numerical result:

1.

**[tex]\( p \rightarrow q \)[/tex]**(If we double the dimensions, the area increases by a factor of 4. True as given)

2.

**[tex]\( \sim q \rightarrow \sim p \)[/tex]**(If the area does not increase by a factor of 4, we did not double the dimensions. True as it is the contrapositive of the original conditional statement)

Therefore, the correct options are indeed:

1. [tex]\( p \rightarrow q \)[/tex]

2. [tex]\( \sim q \rightarrow \sim p \)[/tex]

Thus, the true options correspond to:

**First option:**[tex]\( p \rightarrow q \)[/tex]

**Second option:**[tex]\( \sim q \rightarrow \sim p \)[/tex]

These correspond to options

**1 and 4**in the given multiple-choice question.