Question

Each of these questions is followed by two statements, I and II. Mark the answer as: (a) if the question can be answered with the help of statement I alone. (b) if the question can be answered with the help of statement II alone. (c) if both statement I and statement II are needed to answer the question. (d) if the question cannot be answered even with the help of both the statements.

What is the length of rectangle ABCD?
I. Area of the rectangle is 48 square units.
II. Length of the diagonal is 10 units.

• A. Statement I alone is sufficient
• B. Statement II alone is sufficient
• C. Both statements I and II together are sufficient
• D. Cannot be answered even with both statements

Hint:

Let the sides be \(l\) and \(w\). Statement I gives \(lw\) and Statement II gives \(l^2 + w^2\). Combine them through \((l + w)^2\).

Answer 1

The Correct Option is C

Step 1: Test Statements separately:
Let the sides be \(l\) and \(w\) with \(l > w\). Statement I gives \(lw = 48\): many rectangles fit (for example 6 by 8 or 4 by 12), so I alone is not enough. Statement II gives \(l^2 + w^2 = 100\): many rectangles fit (for example 6 by 8 or 5 by \(\sqrt{75}\)), so II alone is not enough.

Step 2: Combine both:
\((l + w)^2 = l^2 + w^2 + 2lw = 100 + 96 = 196\), so \(l + w = 14\).
\((l - w)^2 = 100 - 96 = 4\), so \(l - w = 2\).
Hence \(l = 8\) and \(w = 6\).

Step 3: Conclude:
The length is 8 units (the longer side). Both statements are needed together.

Final Answer:
The correct option is (3): Both statements I and II together are sufficient.