Question

find the shortest side of $\triangle qrs$.

Explanation:

Step1: Sum angles to 180°

$$62 + (2z+51) + (16z-59) = 180$$

Step2: Simplify the equation

$$62+51-59 + 18z = 180$$
$$54 + 18z = 180$$

Step3: Solve for z

$$18z = 180 - 54$$
$$18z = 126$$
$$z = \frac{126}{18} = 7$$

Step4: Calculate ∠Q

$$\angle Q = 2(7)+51 = 14+51 = 65^\circ$$

Step5: Calculate ∠R

$$\angle R = 16(7)-59 = 112-59 = 53^\circ$$

Step6: Match angles to sides

In a triangle, the shortest side is opposite the smallest angle. The smallest angle is $\angle R = 53^\circ$, which is opposite side $QS$.

Answer:

The shortest side is $\boldsymbol{QS}$