In a rhombus PQRS, each side has length 8 cm and the interior angle ∠PQR is 120°. Find the length (in centimetres) of the diagonal QS of the rhombus.

Introduction:
This question involves properties of a rhombus and the use of trigonometry to find the length of a diagonal. In a rhombus, all four sides are equal, but the diagonals are different and depend on the interior angles. The given information includes the side length and one interior angle, and we are asked to find one of the diagonals, specifically QS. This type of geometry question checks understanding of special quadrilaterals and the relationship between sides, angles, and diagonals.

Given Data / Assumptions:

• PQRS is a rhombus. • Each side of the rhombus is 8 cm. • ∠PQR = 120°. • We are asked to find the length of diagonal QS. • Diagonal QS is taken to be the shorter diagonal, associated with the acute angle of the rhombus.

Concept / Approach:
In a rhombus, opposite angles are equal and adjacent angles are supplementary. Given that ∠PQR = 120°, the adjacent angle is 60°. The diagonals of a rhombus bisect the angles and intersect at right angles. As a result, each diagonal can be related to the side length and the angles using trigonometric identities or using known formulas for diagonals in a rhombus. One useful fact is that for a rhombus of side a and one interior angle θ, the lengths of the diagonals d1 and d2 are connected to the sides and angle by formulas involving 2 ± 2 * cosθ. For the shorter diagonal, in this particular configuration, we obtain a simple relation that gives diagonal QS = side length.

Step-by-Step Solution:
Step 1: Understand the geometry. The rhombus has side 8 cm and angles of 120° and 60°. Diagonal QS is opposite an acute angle in the figure. Step 2: Use the diagonal formula for a rhombus. For a rhombus of side a and angle 120°, one diagonal corresponds to an expression involving 2 + 2 * cos(120°), and the other corresponds to 2 - 2 * cos(120°). Step 3: Recall that cos(120°) = -1/2. Then 2 + 2 * cos(120°) = 2 + 2 * (-1/2) = 2 - 1 = 1. Step 4: The shorter diagonal squared becomes a^2 * 1 = a^2. So, diagonal QS^2 = 8^2 = 64. Step 5: Take the square root. Diagonal QS = 8 cm.

Verification / Alternative check:
Alternatively, you can use vector or coordinate geometry to place the rhombus on a plane, for example by placing PQ along the x-axis and using the 120° angle at Q to locate point R. Coordinate methods again lead to the same diagonal length after computing distances. The consistency of both trigonometric and coordinate approaches supports that the shorter diagonal QS must be 8 cm for the given rhombus side and angle configuration.

Why Other Options Are Wrong:
Option 4√5 cm: This value is approximately 8.94 cm and does not match the specific diagonal length derived from the side and angle data. Option 6 cm: This is too small for a diagonal relative to a side length of 8 cm in this configuration. Option 12 cm: This is too large and would suggest a much more stretched shape than the given angle would allow for the shorter diagonal. Option 10 cm: This is again larger than the side and does not satisfy the derived formula based on the rhombus geometry and angle.

Common Pitfalls:
Students often misinterpret which diagonal is being asked for or assume that both diagonals must be longer than the side. Another mistake is to use the cosine rule incorrectly in a triangle formed by two sides and a diagonal without correctly identifying the included angle. Forgetting the exact value of cos(120°) or mixing up acute and obtuse angles can also lead to errors. Always map the geometry carefully and match it consistently with formulas.

Final Answer:
The length of diagonal QS of the rhombus is 8 cm.