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The following suggestions will be based on trig using the right triangle. The classic problem is finding the height of a flagpole. Almost every school has one. This can first be done by similar triangles. The students will need some type of tripod or pole close to shoulder level where they can set up a protractor to use as a sighting instrument to get an angle for a line of sight. Imagine that it is not possible to measure the distance to the base of the flagpole. Place another student a distance from the base of the protractor that can be measures so that the line of sight from the protractor to the top of the flagpole includes the top of the head of the student. The height of the student can be measured, as can the distance to the student from the protractor. Remember to add the height of the tripod to the height found of the flagpole when doing the problem! If you can measure to the base of the flagpole, then the problem can be done directly. Have you considered having the students mark off the boundaries of a rectangular field in an area that is not rectangular? They can check by measured diagonals and using the Pythagorean theorem to be sure that their angles are correct. If you have developed the Law of Sines, perhaps you can measure the distance across a street by placing two students (A and B) at a known distance on one side of the street. Send a third student (C) across the street. Measure the angle between the line of sight from AB to AC as well as from AB to BC. Once the distance AC or BC is found you can find the altitude to the triangle ABC using one of the sides and the corresponding angle. This problem requires that a "horizontal" angle be found since the problem is in the horizontal plane. The first problem required that a "vertical" angle be found since the problem was in the vertical plane. Finding the height of a smokestack or a tank on the top of your school building is a more difficult problem, since you have to measure two vertical angles. Perhaps the most important part of all of this is the construction of the "tools" to be used. I was once asked by a contractor to figure out the radius of a circle to be used in determining the top of a Palladian window. The top would be a segment of a circle, like when you cut off the crust of a section (piece of pie!). The height of the top was a portion of the radius that bisected the width of the window. The solution involved trig, but is too complicated to type out without reference to a drawing. I tell you this only because these questions are out in the "field" of construction work. I hope that you and your students have a good time doing these problems.
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