Rectangle with diagonal 6 cm, diagonal makes 30° with a side — area In a rectangle, the diagonal has length 6 cm and makes a 30° angle with one side. Find the area of the rectangle.
Aptitude
Height and Distance
Difficulty: Easy
Choose an option
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A9 cm2
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B9 √3 cm2
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C27 cm2
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D36 cm2
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E—
Answer
Correct Answer: 9 √3 cm2
Explanation
Introduction / Context:Knowing the diagonal and its angle with a side determines both side lengths via basic trigonometry, from which the area follows directly.
Given Data / Assumptions:
- Diagonal d = 6 cm.
- Let sides be a (adjacent to the 30°) and b (opposite).
- Angle between diagonal and side a = 30°.
Concept / Approach:tan 30° = b/a = 1/√3 ⇒ b = a/√3. Also, d² = a² + b². Substitute b and solve for a, then compute area A = a·b.
Step-by-Step Solution:
b = a/√3d² = a² + b² = a² + a²/3 = (4/3)a²6² = (4/3)a² ⇒ a² = 27 ⇒ a = 3√3, b = 3Area A = a·b = 3√3 · 3 = 9√3 cm²Verification / Alternative check:Compute diagonal from a and b: √(27 + 9) = √36 = 6 (consistent).
Why Other Options Are Wrong:9 cm² corresponds to b = 3 with a = 3, which would make tan 30° = 1; 27 and 36 cm² are incompatible with d = 6.
Common Pitfalls:Using sin or cos instead of tan to relate a and b at the specified angle.
Final Answer:9 √3 cm2