Each question below is followed by two Statements I and II. You have to determine whether the data given in the statements are sufficient for answering the question. You should use the data and your knowledge of Mathematics to choose between the possible answers. Give answer
What is the perimeter of the square?
I. The measure of one of its sides is given.
II. The measure of its diagonal is given.

Options

A. if the Statement I alone is sufficient to answer the question but the Statement II alone is not sufficient.
B. if the Statement II alone is sufficient to answer the question but the Statement I alone is not sufficient.
C. if both Statement I and II together are needed to answer the question.
D. if either the Statement I alone or Statement II alone is sufficient to answer the question.

Answer

To determine the perimeter of a square, we can analyze the two statements provided:

**Question**: What is the perimeter of the square?

### Step 1: Analyze Statement I
**Statement I**: The measure of one of its sides is given.

- Let the length of one side of the square be denoted as \( s \).
- The formula for the perimeter \( P \) of a square is given by:
\[ P = 4 \times s \]
- Since we have the measure of one side, we can directly calculate the perimeter using this formula.

**Conclusion for Statement I**: This statement alone is sufficient to find the perimeter of the square.

### Step 2: Analyze Statement II
**Statement II**: The measure of its diagonal is given.

- Let the length of the diagonal be denoted as \( d \).
- The relationship between the side length \( s \) of a square and its diagonal \( d \) is given by the formula:
\[ d = s \sqrt{2} \]
- From this, we can express the side length in terms of the diagonal:
\[ s = \frac{d}{\sqrt{2}} \]
- Now, substituting this expression for \( s \) into the perimeter formula:
\[ P = 4 \times s = 4 \times \left(\frac{d}{\sqrt{2}}\right) = \frac{4d}{\sqrt{2}} = 2\sqrt{2} \times d \]
- Thus, we can also calculate the perimeter using the diagonal.

**Conclusion for Statement II**: This statement alone is also sufficient to find the perimeter of the square.

### Final Conclusion
Since both statements I and II are sufficient on their own to determine the perimeter of the square, the correct answer is:

**Answer**: D - Either statement I alone or statement II alone is sufficient to answer the question.

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