Question

1. find the value of x in the figure below.

9 inches
17 inches
18 inches
34 inches

Explanation:

Step1: Find angle at G

The sum of angles in a triangle is \(180^\circ\). So, \(\angle G = 180^\circ - 126^\circ - 27^\circ = 27^\circ\).

Step2: Apply Law of Sines

Law of Sines: \(\frac{x}{\sin I}=\frac{HI}{\sin G}=\frac{GI}{\sin H}\). Since \(\angle G = \angle I = 27^\circ\), the triangle is isosceles with \(x = HI\). Given \(HI = 18\) in? Wait, no, wait. Wait, \(\angle I = 27^\circ\), \(\angle G = 27^\circ\), so sides opposite are equal. Side opposite \(\angle I\) is \(x\) (GH), side opposite \(\angle G\) is \(HI = 18\) in? Wait, no, wait the sides: \(GI = 34\) in (opposite \(\angle H = 126^\circ\)), \(HI = 18\) in (opposite \(\angle G = 27^\circ\)), \(GH = x\) (opposite \(\angle I = 27^\circ\)). Since \(\angle G = \angle I = 27^\circ\), their opposite sides are equal. So \(x = HI = 18\) in? Wait, no, wait \(\angle I = 27^\circ\), opposite side is \(GH = x\); \(\angle G = 27^\circ\), opposite side is \(HI = 18\) in. So \(x = 18\) in? Wait, but let's check again. Wait, angles: \(\angle H = 126^\circ\), \(\angle I = 27^\circ\), so \(\angle G = 180 - 126 - 27 = 27^\circ\). So \(\angle G = \angle I = 27^\circ\), so triangle is isosceles with \(GH = HI\). \(HI\) is 18 in? Wait, the figure: \(HI\) is 18 in, \(GH\) is \(x\), \(GI\) is 34 in. So since \(\angle G = \angle I\), then \(GH = HI\), so \(x = 18\) in? Wait, but the options have 18 inches. Wait, but let's use Law of Sines properly. Law of Sines: \(\frac{x}{\sin I}=\frac{HI}{\sin G}\). Since \(\sin G = \sin I\) (because \(\angle G = \angle I = 27^\circ\)), then \(x = HI\). \(HI\) is 18 in, so \(x = 18\) in. Wait, but wait, maybe I misread the sides. Wait, the side \(HI\) is 18 in, angle at I is 27°, angle at G is 27°, so sides opposite are equal. So \(x\) (opposite angle I) equals \(HI\) (opposite angle G). So \(x = 18\) in.

Answer:

18 inches (corresponding to the option with 18 inches, e.g., if the option is C. 18 inches, then C. 18 inches)