Find the height of a bar, given the lengths of shadows cast by it and another bar

Here is a different visualization. Shadow visualization


3D histograms are evil according to Edward Tufte. Here, they are used to obfuscate information and make this geometry problem harder than it is. Also, as mentioned by @CandiedOrange and @LamarLatrell, the original drawing isn't to scale.

Here's a 3D render with correct heights:

enter image description here

By playing with perspective and point of view, you can seamlessly merge lengths that appear on distinct axes. It might give you the wrong impression that you could simply add those lengths.

from above

But if you select the correct perspective, the problem becomes much clearer.

enter image description here

enter image description here


There are three similar right triangles here and the ratios of their catheti (or legs) are all equal: $$\frac 23=\frac{4}{x}=\frac{h}{6+x} \implies x=6,\quad h=\frac{2(6+x)}{3}=8$$ ($x$ is the an imaginary cathetus which goes beyond the blue wall).

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Geometry

Soft Question

Algebra Precalculus

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