Consider a triangle with $x \in (0,6)$ and $h \in (0, 8)$:
Question: Often times in questions related to volumes I have leveraged the property that $x$ and $h$ are linearly dependent to get the relationship $h = 8 - \frac{4}{3}x$. How can we see this visually?
(Edit: Something a little more than similar triangles, to intuitively see how $x$ and $h$ move together in a linear way)
Linear algebra could help visualising why $ h$ and $x $ are linearly dependent as you can assume left adjacent sides as $X'-H'$ axis with angle $\color{green}{\theta}$, which are transformed from rectangular coordinate $X-H$ axis by using matrix $$ \begin{bmatrix}1&0\\0&\frac{1}{sin\theta}\end{bmatrix}$$ and hence in oblique coordination we have $$\begin{bmatrix}x'\\h'\end{bmatrix}=\begin{bmatrix}x\\\frac{h}{sin\theta}\end{bmatrix}$$ where $\theta $ is the angle between left adjacent sides of triangle.Now the third side of triangle, which is a segment of line $ \color{green}{(} $all points on line are linearly dependent $\color{green}{)}$, can be expressed as linear combination of $x'$and $h'$ and hence of $x$ and $h$.