Rhombus with side 8 cm and ∠PQR = 120°\nIf rhombus PQRS has each side 8 cm and ∠PQR = 120°, find the length of diagonal QS (in cm).

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:

The diagonal opposite the obtuse angle is shorter. With θ = 120°, cosθ = −1/2.

Step-by-Step Solution:

Verification / 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