Final answer:
The question applies geometry to calculate the size of the Moon as observed from Earth and to determine the third side of a land parcel. It includes the use of congruent triangles and vectors in both celestial and terrestrial contexts.
Step-by-step explanation:
Understanding Geometry in Real-World Contexts
The question involves applying geometry to understand and solve real-world problems. In particular, the geometry of celestial observations and land measurements are the focus. The problem relating to the measurement of the Moon involves congruent triangles and visual angles to deduce that AC equals three times the radius of the celestial body (R), while the land fencing problem requires knowledge of vectors to calculate the third side of a land parcel.
Calculating the third side of a triangle in a land fencing context can be done by understanding that the vectors representing two sides of the triangle can form a parallelogram, where the diagonal represents the third side. On the subject of observing the moon's width from Earth, the problem illustrates the use of congruent triangles to establish a relationship between the size of an object at a distance and the angles subtended by that object.
The width of the moon as seen from point H is KD = x and the angle KHD is 0.5 degrees. By extending the line AD a further distance R to point F, we can prove that the triangles HKD and KFD are congruent to the two shaded triangles.
Based on this congruency, we can determine that AC = 3R and AB = 3x, using simple proportions.
Apart from applications in astronomical observations, understanding angles and triangles is fundamental in many areas of study, including surveying and navigation.