Square from Diagonal — Compute Area: If the diagonal AC of square ABCD is 5.2 cm, find the area of the square ABCD (in square centimetres).
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A15.12 sq.cm
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B13.52 sq.cm
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C12.62 sq.cm
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D10 sq.cm
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E14.04 sq.cm
Answer
Correct Answer: 13.52 sq.cm
Explanation
Introduction / Context:For any square, the diagonal relates to the side by d = s * √2. Once the side is known, area follows directly from s^2. This question checks your ability to manipulate the diagonal–side relationship and maintain consistent units (centimetres).
Given Data / Assumptions:
- Diagonal d = 5.2 cm
- Square property: d = s * √2
- Area of square: A = s^2
- All measures in centimetres
Concept / Approach:The diagonal of a square is the hypotenuse of an isosceles right triangle with legs equal to the side length s. Therefore, s = d / √2. After computing s, square it to obtain the area in square centimetres. Only simple arithmetic and the √2 relation are required—no advanced trigonometry.
Step-by-Step Solution:
Compute side: s = d / √2 = 5.2 / √2.Since √2 ≈ 1.41421356, s ≈ 5.2 / 1.41421356 ≈ 3.676955 cm.Area: A = s^2 ≈ (3.676955)^2 ≈ 13.52 sq.cm (rounded to two decimals).Verification / Alternative check:
Reverse check: If A = 13.52 ⇒ s ≈ √13.52 ≈ 3.6769 cm; diagonal = s * √2 ≈ 3.6769 * 1.4142 ≈ 5.20 cm, which matches the given 5.2 cm within rounding.Why Other Options Are Wrong:
- 15.12 and 12.62 result from rounding √2 or squaring errors.
- 10 sq.cm underestimates, implying a side near 3.16 cm, which would give a diagonal far below 5.2 cm.
- 14.04 sq.cm is a near miss from using √2 ≈ 1.45 or other approximations.
Common Pitfalls:
- Using area = (diagonal)^2 instead of dividing by 2; remember A = d^2 / 2 for a square.
- Mixing units or rounding too early; keep sufficient precision until the final step.
Final Answer:13.52 sq.cm.