Given problem consists of a problem followed by two statements. Decide whether…

2023

Given problem consists of a problem followed by two statements. Decide whether the data in the statements are sufficient to answer the question: What is the length of the diagonal of rectangle ABCD?

1. The perimeter of the rectangle is 16.

2. The area of the rectangle is 16.

  1. A.

    both statements taken together are sufficient to answer the question, but neither statement alone is sufficient

  2. B.

    statement 2 alone is sufficient, but statement 1 alone is not sufficient to answer the question

  3. C.

    each statement alone is sufficient

  4. D.

    statement 1 alone is sufficient, but statement 2 alone is not sufficient to answer the question

Show answer & explanation

Correct answer: A

Concept: In Data Sufficiency, a statement is sufficient only if it fixes ONE unique value. For a rectangle with sides l and w, the perimeter gives l + w, the area gives lw, and the diagonal is d = √(l2 + w2). The identity l2 + w2 = (l + w)2 − 2lw lets the diagonal be found once BOTH l + w and lw are known.

Statement 1 alone (perimeter = 16): l + w = 8. Many rectangles satisfy this — e.g. l = 1, w = 7 gives diagonal √50, while l = 3, w = 5 gives diagonal √34. Different pairs give different diagonals, so statement 1 alone is not sufficient.

Statement 2 alone (area = 16): lw = 16. Many rectangles satisfy this too — e.g. l = 1, w = 16 gives diagonal √257, while l = 2, w = 8 gives diagonal √68. Different pairs again give different diagonals, so statement 2 alone is not sufficient.

Combining both statements:

  1. From l + w = 8 and lw = 16, l and w are the roots of x2 − 8x + 16 = 0.

  2. This factors as (x − 4)2 = 0, so the only solution is l = w = 4.

  3. Then l2 + w2 = (l + w)2 − 2lw = 82 − 2(16) = 32.

  4. So the diagonal d = √32 = 4√2 — a single, unique value.

Cross-check: perimeter = 2(4 + 4) = 16 and area = 4 × 4 = 16, confirming l = w = 4 satisfies both statements. Since the resulting quadratic has a zero discriminant, this is the only pair satisfying both conditions together — so combining the statements pins down a unique diagonal even though each statement alone does not.

Both statements taken together are sufficient to determine the diagonal, but neither statement alone is sufficient.

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