Question

what is the value of w?

Explanation:

Step1: Identify the triangle type

Since two angles are marked as equal (the red - arc angles at \(F\) and \(G\)), \(\triangle FGH\) is an isosceles triangle. In an isosceles triangle, the sides opposite the equal angles are equal.

Step2: Apply the property of isosceles triangle

The side opposite the angle at \(F\) is \(HG = w\), and the side opposite the angle at \(G\) is \(HF=12\). Wait, no, correction: in a triangle, if \(\angle F=\angle G\), then the sides opposite them are equal. The side opposite \(\angle F\) is \(HG\) and the side opposite \(\angle G\) is \(HF\). Wait, no, another correction: if \(\angle F=\angle G\), then \(HG = HF\). Wait, no, the side - angle relationship: in \(\triangle FGH\), if \(\angle F=\angle G\), then \(HG = HF\). But wait, no, the side opposite \(\angle F\) is \(HG\) and the side opposite \(\angle G\) is \(HF\). Wait, no, the Law of Sines: \(\frac{HG}{\sin\angle F}=\frac{HF}{\sin\angle G}\). Since \(\angle F = \angle G\), then \(HG=HF\). But wait, no, looking at the side lengths: if \(\angle F=\angle G\), then \(HG = HF\). Wait, no, the side adjacent to \(\angle F\) is \(FG = 7\) and \(FH = 12\), no, wait, no, the problem is missing information. Wait, no, re - checking: if two angles of a triangle are equal, then the sides opposite those angles are equal. In \(\triangle FGH\), if \(\angle F=\angle G\), then \(HG=HF\). But \(HF = 12\), so \(w = 12\).

Answer:

\(12\)