ke use of the cosine rule to determine the length of PQ

It sounds like you’re being asked to find the length of side PQ in a triangle by applying the cosine rule. (Note: Without a specific diagram or numerical details, I’ll explain the general procedure.) Suppose you have triangle PQR and you wish to find the length of side PQ. The cosine rule (or law of cosines) in any triangle tells you that if you know two sides and the included angle (or three sides), you can find the missing side. In its most common form, the cosine rule is expressed as follows:   For any triangle with sides a, b, and c, where side a is opposite angle A:    a² = b² + c² – 2·b·c·cos A In triangle PQR, let’s assume that:  • Side PQ is opposite a certain angle (say, angle R)  • Sides PR and QR are known Then by the cosine rule we have:   (PQ)² = (PR)² + (QR)² – 2·(PR)·(QR)·cos(angle R) To solve for PQ:  1. Plug in the known values for PR, QR, and the measure of angle R.  2. Compute the right-hand side.  3. Finally, take the square root of the result to get PQ:   PQ = √[(PR)² + (QR)² – 2·(PR)·(QR)·cos(angle R)] If instead you know a different angle (for example, angle Q), make sure you identify the angle that is opposite the side you want to find. For instance, if angle Q is opposite side PR, then you’d write the cosine rule accordingly for that side. The key point is to match the unknown side with the angle directly opposite it. In summary, the steps are:  • Identify the side you need (PQ) and the angle opposite it.  • Use the cosine rule:   PQ² = (side adjacent 1)² + (side adjacent 2)² – 2·(side adjacent 1)·(side adjacent 2)·cos(opposite angle)  • Solve for PQ by taking the square root of PQ². Once you have the numerical values, substitute them into the formula and calculate PQ.