Question

What is the area of a square? Statement I: The perimeter of the square is 20 cm. Statement II: The diagonal of the square is \(\sqrt{50}\) cm.

• A. Statement I alone is sufficient, but Statement II alone is not sufficient.
• B. Statement II alone is sufficient, but Statement I alone is not sufficient.
• C. Both statements together are sufficient, but neither statement alone is sufficient.
• D. Each statement alone is sufficient.

Hint:

Recall that a square's area can be written directly in terms of its perimeter (Area = P squared over 16) or directly in terms of its diagonal (Area = d squared over 2), without solving for the side first.

Answer 1

The Correct Option is D

Step 1: Instead of first solving for the side length and then squaring it, apply direct area formulas that use the perimeter or diagonal without an intermediate side-length step.
Step 2: For a square with perimeter $P$, the side is $\frac{P}{4}$, so the area formula directly in terms of perimeter is $\text{Area}=\left(\frac{P}{4}\right)^2=\frac{P^2}{16}$.
Step 3: Statement I: $P=20$ cm, so Area $=\frac{20^2}{16}=\frac{400}{16}=25$ cm$^2$. This is a single unique value, so Statement I alone IS sufficient.
Step 4: For a square with diagonal $d$, the direct area formula is $\text{Area}=\frac{d^2}{2}$ (since diagonal $d=s\sqrt2$, so $d^2=2s^2$, giving $s^2=\frac{d^2}{2}$, and $s^2$ is exactly the area).
Step 5: Statement II: $d=\sqrt{50}$ cm, so Area $=\frac{(\sqrt{50})^2}{2}=\frac{50}{2}=25$ cm$^2$. This is also a single unique value, so Statement II alone IS sufficient. Since each statement independently pins down the area, each statement alone is sufficient. \[\boxed{\text{Each statement alone is sufficient.}}\] Correct option: (4) Each statement alone is sufficient.