Question

list the side lengths of \\( \delta wxy \\) in order from shortest to longest.

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Explanation:

Step1: Compare the angles

In a triangle, the larger the angle, the longer the side opposite it.
We have angles \(62s\), \(43s\), \(s + 74^{\circ}\).
Let's first find the value of \(s\) using the angle - sum property of a triangle (\(A+B + C=180^{\circ}\)).
\(62s+43s+(s + 74)=180\)
\(62s+43s+s+74 = 180\)
\(106s=180 - 74\)
\(106s = 106\)
\(s = 1\)
So the angles are \(\angle X=62^{\circ}\), \(\angle Y = 43^{\circ}\), \(\angle W=1 + 74=75^{\circ}\)

Step2: Order the angles

Order of angles from smallest to largest: \(\angle Y(43^{\circ})<\angle X(62^{\circ})<\angle W(75^{\circ})\)

Step3: Order the sides

The side opposite \(\angle Y\) is \(WX\), the side opposite \(\angle X\) is \(WY\), and the side opposite \(\angle W\) is \(XY\)
Since in a triangle, if \(\angle A<\angle B<\angle C\), then \(a < b < c\) (where \(a\) is the side opposite \(\angle A\), \(b\) is the side opposite \(\angle B\), \(c\) is the side opposite \(\angle C\))

Answer:

\(WX