Question

find the shortest side of $\triangle rst$.

Explanation:

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

The sum of the interior angles of a triangle is \(180^\circ\). So, we set up the equation: \((u + 59) + (u + 64) + 55u = 180\).

Step2: Simplify the equation

Combine like terms: \(u + 59 + u + 64 + 55u = (1 + 1 + 55)u + (59 + 64) = 57u + 123 = 180\).

Step3: Solve for \(u\)

Subtract 123 from both sides: \(57u = 180 - 123 = 57\). Then divide both sides by 57: \(u = \frac{57}{57} = 1\).

Step4: Find each angle

• Angle \(R\): \(u + 59 = 1 + 59 = 60^\circ\)
• Angle \(S\): \(u + 64 = 1 + 64 = 65^\circ\)
• Angle \(T\): \(55u = 55\times1 = 55^\circ\)

Step5: Determine the shortest side

In a triangle, the shortest side is opposite the smallest angle. The smallest angle is \(55^\circ\) (angle \(T\)), so the side opposite angle \(T\) is \(RS\).

Answer:

\(RS\)