Question

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

Explanation:

Step1: Sum angles to find $b$

The sum of angles in a triangle is $180^\circ$.
$$31b + 24b + (b+68) = 180$$
$$56b + 68 = 180$$
$$56b = 112$$
$$b = 2$$

Step2: Calculate each angle

Substitute $b=2$ into each angle expression.

• $\angle U = 31(2) = 62^\circ$
• $\angle T = 24(2) = 48^\circ$
• $\angle V = 2 + 68 = 70^\circ$

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:

• Smallest angle $\angle T=48^\circ$ → opposite side $UV$
• Middle angle $\angle U=62^\circ$ → opposite side $TV$
• Largest angle $\angle V=70^\circ$ → opposite side $TU$

Answer:

$UV < TV < TU$