explain how one can find the breadth of the river without crossing it
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Measure the distance along a line from a point directly opposite any landmark on the other bank to a point some distance upstream or downstream. Then measure the angle between that line and a sighting toward the original landmark. The width of the river is the distance between the two points on your side of the river times the tangent of the angle.
This can also be solved by using any two points on your bank of the river. Measure the distance between them as a baseline and then measure the angles between the baseline and a line of sight to any landmark point on the far bank.
The sum of those two angles will give you the third angle of the triangle formed by the three points. You have one side and all three angles, so you can find the other sides using the law of sines for triangles.
Then, the width of the river (from the landmark on the far side perpendicularly to the baseline) is the altitude of the triangle. There are a number of ways to find the altitude given all three sides. Heron's formula will give the area, and from that and the baseline, the height can be found using A = bh/2, solved for h.
This second method is essentially how astronomers calculate the distance from the earth to nearby stars. Angles are measured from the earth to the star 1/2 year apart using a diameter of the earth's orbit as the baseline (actually, not a diameter since the orbit is elliptical rather than circular).
This can also be solved by using any two points on your bank of the river. Measure the distance between them as a baseline and then measure the angles between the baseline and a line of sight to any landmark point on the far bank.
The sum of those two angles will give you the third angle of the triangle formed by the three points. You have one side and all three angles, so you can find the other sides using the law of sines for triangles.
Then, the width of the river (from the landmark on the far side perpendicularly to the baseline) is the altitude of the triangle. There are a number of ways to find the altitude given all three sides. Heron's formula will give the area, and from that and the baseline, the height can be found using A = bh/2, solved for h.
This second method is essentially how astronomers calculate the distance from the earth to nearby stars. Angles are measured from the earth to the star 1/2 year apart using a diameter of the earth's orbit as the baseline (actually, not a diameter since the orbit is elliptical rather than circular).
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