Explanation:
The key fact about similar triangles is corresponding parts are proportional.
That means the ratios between corresponding parts are the same. In this context, the parts can be any linear measure of the triangle, including sides, sums of sides, altitudes, and so on.
In the context of this problem, the similarity of the triangles means that short sides have the same ratio as long sides:
[tex]\dfrac{\text{stake 3 to stake 1}}{\text{stake 3 to tree 1}}=\dfrac{\text{stake 1 to stake 2}}{\text{tree 1 to tree 2}}\\\\\dfrac{30\,ft}{30\,ft+\text{river width}}=\dfrac{35\,ft}{140\,ft}[/tex]
Of course, any given proportion can be written 4 ways. It often works well to choose one that puts the unknown variable in the numerator. Those ways are ...
- a/b = c/d
- b/a = d/c
- a/c = b/d
- c/a = d/b
Here, that means the above proportion can be rewritten as ...
[tex]\dfrac{30\,ft+\text{river width}}{30\,ft}=\dfrac{140\,ft}{35\,ft}[/tex]
This is easily solved by multiplying by 30 ft to get ...
[tex]30\,ft+\text{river width}=120\,ft\\\\\text{river width}=90\,ft[/tex]
_____
(A) We don't know what the diagram on the previous page shows, so we don't know what equation or solution it might give rise to.
(B) Perimeter is a linear measure of a triangle. The perimeters of similar triangles will have the same ratio as any pair of corresponding parts. The triangles shown on this page have long sides in the ratio (140 ft)/(35 ft) = 4, so the perimeter of the larger triangle will be 4 times that of the smaller one.
(C) Diagram not provided here.
(D) I have been on a surveying crew, and I can tell you it is exceedingly difficult to make the same measurement two different ways and get the same result. Reality aside, the geometry is quite reliable, so, mathematically, the results will be the same if "stake 1" is located in the same place by the two different crews.
