Question

list the side lengths of $\triangle qrs$ in order from shortest to longest.

Explanation:

Step1: Sum angles to find $b$

The sum of angles in a triangle is $180^\circ$.
$$22b + (54b - 47) + (60b - 45) = 180$$
$$22b + 54b + 60b - 47 - 45 = 180$$
$$136b - 92 = 180$$
$$136b = 180 + 92$$
$$136b = 272$$
$$b = \frac{272}{136} = 2$$

Step2: Calculate each angle

• $\angle Q = 22b = 22(2) = 44^\circ$
• $\angle S = 54b - 47 = 54(2) - 47 = 108 - 47 = 61^\circ$
• $\angle R = 60b - 45 = 60(2) - 45 = 120 - 45 = 75^\circ$

Step3: Match sides to opposite angles

In a triangle, the shortest side is opposite the smallest angle, and the longest side is opposite the largest angle:

• Side opposite $\angle Q$: $RS$
• Side opposite $\angle S$: $QR$
• Side opposite $\angle R$: $QS$

Order of angles (smallest to largest): $\angle Q < \angle S < \angle R$
Corresponding side order: $RS < QR < QS$

Answer:

$RS < QR < QS$