The contra-positive of the following statement, "If the side of a square doubles, then its area increases four times", is

Options

A. if the area of a square increases four times, then its side is not doubled
B. if the area of a square.increases four times, then its side is doubled
C. if the area of a square does not increase fourtimes, then its side is not doubled
D. if the side of a square is not doubled, then its area does not increase four times

Answer

To find the contrapositive of the statement "If the side of a square doubles, then its area increases four times," we will follow these steps:

### Step 1: Identify the statements
Let:
- \( p \): The side of a square doubles.
- \( q \): The area of the square increases four times.

The original statement can be expressed as:
\[ p \implies q \]

### Step 2: Write the negations of the statements
Now we need to find the negations of \( p \) and \( q \):
- The negation of \( q \) (area does not increase four times) is:
\[ \neg q: \text{The area of the square does not increase four times.} \]

- The negation of \( p \) (the side of the square does not double) is:
\[ \neg p: \text{The side of the square does not double.} \]

### Step 3: Formulate the contrapositive
The contrapositive of the implication \( p \implies q \) is given by:
\[ \neg q \implies \neg p \]

Substituting the negations we found:
\[ \text{If the area of the square does not increase four times, then the side of the square does not double.} \]

### Final Answer
Thus, the contrapositive of the statement "If the side of a square doubles, then its area increases four times" is:
**If the area of the square does not increase four times, then the side of the square does not double.**

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