In the angle shown below, as the distance (represented by a, x) from the vertex of the angle changes, the height (represented by b, h) also changes.

Given that $h \propto x$ , find the constant of proportionality.

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Solution

Let us consider a point A on the slanted line at a distance of $a$ units from O and $b$ units from the baseline.

Let C be another such point which is $x$ units from O and $h$ units from E.

Then $△ O A D \sim △ O C E$ (AA similarity)

$⟹ \frac{O A}{O C} = \frac{A D}{C E}$

i.e., $\frac{a}{x} = \frac{b}{h}$

$⟹ h = x \times \frac{a}{b}$

$⟹ h \propto x$

Thus, the constant of proportionality is $\frac{a}{b}$

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