Superlesson
Project 7-3
Geometry
Chapter 7, Similarity
Long ago, surveyors used various tools and their knowledge of trigonometry to calculate distances which they couldn't calculate directly. Presently, the tools surveyors use measure distances much more precisely, but trigonometry is still used to calculate distances indirectly.
Part A, Trigonometric Ratios
1. Look at the Uses of Trigonometry Web site to see one simple application of trigonometry to measuring geographical distances.
a. Read through the given explanation. Now, suppose you used the same method to figure out the distance across another river. You set up your survey post directly across from a tree on the riverbank. You head downstream 50 meters, and take a sighting of the tree. Your sighting angle is 62°. Draw a sketch of this situation and calculate the distance across the river.
b. In some situations, you can directly measure the distance across a river by stringing a rope across the river. What are some situations in which you would need to perform this measurement indirectly as shown above?
Part B, Angles of Elevation and Depression
2. Learn alternate methods of measuring heights.
a. The method above for making an indirect measurement is fairly simple, but requires the surveyor to set up a survey post on one edge of the object to be measured. This is not always possible. However, there is another technique which allows surveyors to measure heights indirectly by setting up two survey posts in convenient spots. Go to the Distant mountain Web site to see an example of the use of this method. Look through the "Hints" and the "Solution" until you feel comfortable with the calculations.
b. Why would it be impossible to use the method in 1a to measure the height of South Sister?
3. Geometrical quadrants and measurements.
a. Flemish mathematicians of the sixteenth century used a tool called a geometrical quadrant to make indirect measurements using the method illustrated in 2a. Go to the Figure 19 Web site to see a sketch of a geometrical quadrant in use.
b. Suppose you used a geometric quadrant to measure the height of a mountain. Referring to the notation in the illustration on the internet site given in 3a, you sight the top of the mountain at two positions, points O and F. You also measure the distance between points O and I to be 100 feet.
