Question

find the longest side of $\triangle pqr$.

Explanation:

Step1: Sum of angles in a triangle is \(180^\circ\)

The sum of the interior angles of a triangle is \(180^\circ\). So, we can write the equation: \((u + 54) + (u + 61) + 3u = 180\)

Step2: Simplify the left - hand side

Combine like terms: \(u+u + 3u+54 + 61=180\), which simplifies to \(5u+115 = 180\)

Step3: Solve for \(u\)

Subtract 115 from both sides: \(5u=180 - 115=65\)
Divide both sides by 5: \(u=\frac{65}{5}=13\)

Step4: Find each angle

• Angle at \(P\): \(u + 54=13 + 54 = 67^\circ\)
• Angle at \(R\): \(u + 61=13+61 = 74^\circ\)
• Angle at \(Q\): \(3u=3\times13 = 39^\circ\)

Step5: Determine the longest side

In a triangle, the longest side is opposite the largest angle. The largest angle is \(74^\circ\) (at \(R\)), so the side opposite angle \(R\) is \(PQ\).

Answer:

\(PQ\)