Rhombus with side 8 cm and ∠PQR = 120° If rhombus PQRS has each side 8 cm and ∠PQR = 120°, find the length of diagonal QS (in cm).
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A4√5
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B6
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C8
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D12
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ENone of these
Answer
Correct Answer: 8
Explanation
Introduction / Context:In a rhombus (all sides equal), the diagonals depend on the included angle between sides. There are compact formulas for the diagonals in terms of side s and angle θ between adjacent sides.
Given Data / Assumptions:
- Side s = 8 cm
- Angle at Q (∠PQR) = 120°
- Diagonals are PR and QS; one is shorter, one longer.
Concept / Approach:For a rhombus with side s and included angle θ, diagonals are:
d1 = s * sqrt(2 + 2*cosθ) and d2 = s * sqrt(2 − 2*cosθ)The diagonal opposite the obtuse angle is shorter. With θ = 120°, cosθ = −1/2.
Step-by-Step Solution:
d_short = s * sqrt(2 + 2*(-1/2)) = s * sqrt(1) = 8 d_long = s * sqrt(2 − 2*(-1/2)) = s * sqrt(3) = 8√3 QS corresponds to the shorter diagonal for the given vertex angle ⇒ 8 cmVerification / Alternative check:In a rhombus, the diagonal through the obtuse angle is shorter; 8√3 (~13.86) and 8 are consistent magnitudes for the pair.
Why Other Options Are Wrong:4√5 (~8.94), 6, and 12 do not match either diagonal as per the exact formulas for θ = 120° and s = 8 cm.
Common Pitfalls:Mixing up which diagonal is shorter/longer at an obtuse angle or using law of cosines on the wrong triangle can cause confusion.
Final Answer:8