The stars
The methods of triangulation which we used to measure the width of the lecture theatre can be used for much larger distances. Accurate maps and the elevations of mountains such as Everest are determined by triangulation. The main influence on the accuracy of the result is the length of the baseline, which was the distance CF on our worksheet.
In fact, triangulation can even be used to find the distance to stars. The diameter of the earth's orbit can be used as a baseline. One simply has to wait six months between two measurements. The greatest distance measurable by this method from earth is about 1018 meters. Because of its greater clarity, the measurements from the Hubble space telescope extend this range considerably.
Variations on the theme
Problems which seem different from an example you've had may, in fact , be just a variation on the basic problem. The method used to solve the variation is just a small modification on the method used in the basic problem. Once you understand the basic principle you shouldn't need an example in order to solve every type of problem which is based on the principle. Here are some examples of variations on the triangulation theme.
Change orientation or symbols
Flipping the geometry of the problem upside down or backwards leaves the problem essentially unchanged. Likewise labelling the lines differently, e.g, using small letters for the line instead of the large letters of the endpoints, is just a small change. Such variations shouldn't require another example for you to see how to solve them. Just make the necessary modifications on the basic example.Likewise, if one knows h and needs to find d, just solve the equality for d instead of h.
You know the angle instead of the ratio of sides
If one knows the angle, one can use trigonometric functions, which are just shorthand for ratios of the sides of the triangle. Here the ratio r/h is just the tangent of the angle at the lower left-hand tip.
One similar triangle is outside the other one.
Here we use the fact that when two lines cross the opposite angles are equal. So if both triangles are right triangles, they are similar too.
The distances represent other quantities
We might let a distance on one axis represent a quantity other than distance which is proportional to another quantity which is represented by a distance on the other axis. For example, mass of a material is proportional to its volume.
