Final answer:
The unknown measures of triangle ABC similar to triangle JGH can be found using the properties of congruent triangles and simple proportions. For vector problems, the geometric construction method and the Pythagorean theorem can be used to find resultant magnitudes and directions.
Step-by-step explanation:
To find the unknown measures of triangle ABC similar to triangle JGH, given that the width of the Moon as seen from point H is KD = x and the angle KHD is shaded and equals 0.5 degrees, we'll make use of geometric relationships and properties of similar triangles. First, it is established that triangles HKD and KFD are congruent and that triangles GFC and AHD are also congruent to the shaded triangles. Based on this information and geometric properties, we conclude that AC equals 3R, which is thrice the distance R mentioned in the problem. By comparing the similar triangles and using simple proportions, we find that AB equals 3x.
When dealing with vector sums, a geometric construction of the resultant vector can be achieved by adding and subtracting vectors graphically. In the problem stated, the resultant vector R was calculated to be 5.8 cm with an angular direction very close to the horizontal at 0°. Similarly, the vector difference D was found to be 16.2 cm at an angle of 49.3° to the horizontal.
Additionally, the Pythagorean theorem, which states that a² + b² = c², can be applied to calculate the lengths of sides in right triangles or resultants of vector components, as shown in Figure 5.7 for vectors F1, F2, and Ftot.