Final Answer:
1. The bisector of angle FGH is constructed.
2. The point where the bisector intersects FH is labeled as X.
3. Measurements: FX = 3√3 cm, XH = 3 cm.
Step-by-step explanation:
In triangle FGH, the angle bisector theorem states that the angle bisector of a triangle divides the opposite side in such a way that the ratio of the two line segments is equal to the ratio of the other two sides of the triangle. Applying this theorem, we can find that FX/XH is equal to FG/GH. Given that HG = 6 cm, we can set up the equation: FX/XH = FG/HG. Since FHG is a 30-60-90 triangle (as FHG = 30°), we can find FG using the side lengths in a 30-60-90 triangle. The ratio of the sides in a 30-60-90 triangle is 1:√3:2, so FG = HG/2 = 6/2 = 3 cm. Therefore, FX/XH = 3/6, and solving for FX and XH, we find FX = 3√3 cm and XH = 3 cm.
To construct the bisector, use a compass to draw arcs from F and G intersecting at point Y. Then, draw a line from H through Y, and where this line intersects FG, label it as point X. FX and XH can be measured directly using a ruler.
In conclusion, the measurements of FX and XH are 3√3 cm and 3 cm, respectively, and the construction of the bisector involves the use of the angle bisector theorem and basic properties of a 30-60-90 triangle.