If length of each side of a rhombus  PQRS  is  8cm  and  ∠PQR  =  120∘ , then what is the length (in  cm  ) of QS?

To find the length of QS, we can use the properties of a rhombus.

Step 1: Recall that in a rhombus, all sides are equal in length. Given that each side of the rhombus PQRS is 8cm, we know that PQ = QR = RS = SP = 8cm.

Step 2: We are given that ∠PQR = 120°. In a rhombus, opposite angles are equal. Therefore, ∠QRS = ∠PQR = 120°.

Step 3: Since we have an angle and a side length, we can use the Law of Cosines to find the length of QS. The Law of Cosines states that in a triangle, the square of one side is equal to the sum of the squares of the other two sides minus twice the product of the lengths of those sides multiplied by the cosine of the included angle.

Applying the Law of Cosines to triangle QRS, we have: QS^2 = QR^2 + RS^2 - 2(QR)(RS)cos(∠QRS)

Substituting the given values, we get: QS^2 = 8^2 + 8^2 - 2(8)(8)cos(120°)

Simplifying further: QS^2 = 64 + 64 - 128cos(120°)

Step 4: Now, we need to find the value of cos(120°). Since cos(120°) = -1/2, we can substitute this value into the equation: QS^2 = 64 + 64 - 128(-1/2)

Simplifying further: QS^2 = 128 + 64

Step 5: Adding the values: QS^2 = 192

Step 6: Taking the square root of both sides to find QS: QS = √192

Step 7: Simplifying the square root: QS ≈ 13.86 cm

Therefore, the length of QS is approximately 13.86 cm.