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

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.