Question

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

Explanation:

Step1: Set up angle sum equation

The sum of interior angles in a triangle is $180^\circ$, so:
$$(25z+19) + 61z + 75z = 180$$

Step2: Simplify and solve for z

Combine like terms and isolate z:
$$25z+61z+75z+19 = 180$$
$$161z = 180 - 19$$
$$161z = 161$$
$$z = 1$$

Step3: Calculate each angle

Substitute $z=1$ into each angle expression:

• $\angle X = 25(1)+19 = 44^\circ$
• $\angle Y = 61(1) = 61^\circ$
• $\angle W = 75(1) = 75^\circ$

Step4: 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 $\angle X$: $WY$
• Side opposite $\angle Y$: $XW$
• Side opposite $\angle W$: $XY$

Order angles: $44^\circ < 61^\circ < 75^\circ$, so sides follow the same order relative to their opposite angles.

Answer:

$WY < XY < XW$