Question

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

• A. The question can be answered with the help of statement I alone.
• B. The question can be answered with the help of statement II alone.
• C. Both statement I and statement II are needed to answer the question.
• D. The question cannot be answered even with the help of both the statements.

Hint:

In rectangle problems, area + diagonal length uniquely determines dimensions.

Answer 1

The Correct Option is C

Step 1: Analysing Statement I Area = $l \times b = 48$. Infinite pairs $(l, b)$ satisfy this. Not sufficient. Step 2: Analysing Statement II Diagonal = $\sqrt{l^2 + b^2} = 10 \Rightarrow l^2 + b^2 = 100$. Infinite $(l, b)$ pairs possible. Not sufficient. Step 3: Combining Statements From I: $l \times b = 48$, from II: $l^2 + b^2 = 100$. We can solve: $(l + b)^2 = l^2 + b^2 + 2lb = 100 + 96 = 196 \Rightarrow l + b = 14$. Solving $l + b = 14$, $lb = 48$ gives quadratic $x^2 - 14x + 48 = 0$, roots $x = 8, 6$. Length = $8$ units. Step 4: Conclusion Both statements are require(d)