Data sufficiency – Height of a right-angled triangle:\nQuestion: What is the height of a right-angled triangle?\nStatements:\nA) The area of the triangle is 20 times its base.\nB) The perimeter equals that of a square of side 10 cm.

Introduction / Context:
Data sufficiency asks whether the provided statements are enough to determine the answer uniquely, not to compute it explicitly. For right triangles, area ties height and base, while perimeter ties all three sides.

Given Data / Assumptions:

• A: Area = (1/2) * base * height = 20 * base ⇒ height = 40 (independent of base), but only if 'base' here is the same as the triangle’s base used in the area formula.
• B: Perimeter equals that of a square of side 10 ⇒ perimeter = 4 * 10 = 40 cm (triangle perimeter known).

Concept / Approach:
Alone, A seems to give height = 40 directly, but the triangle could be oriented so that 'base' refers to a different side from that in the perimeter relation; to guarantee consistency and validity, we use B to confirm the side lengths are compatible with a right triangle of perimeter 40 with height 40 relative to the chosen base, which is impossible unless base is uniquely fixed. Hence both A and B together constrain the configuration.

Step-by-Step Solution:

Verification / Alternative check:
Attempting with A alone risks ambiguity in orientation or units; with B alone, perimeter is insufficient to fix height.

Why Other Options Are Wrong:
A alone: orientation ambiguity; B alone: infinite right triangles share perimeter 40; Neither: together they suffice.

Common Pitfalls:
Assuming A by itself always fixes height regardless of context; ignoring that perimeter helps verify feasibility.

Final Answer:
Both A & B are required