A half-full pipe
Chord and Distance from Centre
Look down into a stormwater pipe. The water's surface is a straight line. As the water rises, that line first widens, then narrows again. Where is it widest, and why exactly there?
Three ways to see it
The pipe is round, the water surface is flat. Where they meet is a line segment — a chord. As the level changes the chord's length changes, but the pipe's radius doesn't. So the width alone is a measure of the depth.
Switch layers — the scene stays put
What's really going on
The real idea is that the perpendicular BISECTS the chord. That isn't a rule to memorise, it's a consequence of symmetry: both ends of the chord are the same distance from the centre (both are radii), so the triangle is isosceles, so the perpendicular from the apex bisects the base. Once you see that, chord problems stop being about measuring and collapse into one right triangle. And note: r² = d² + (a/2)² is a single equation in three quantities — whichever one is asked, it is the same equation. In the pipe that means: whoever measures the surface width knows the water's depth without ever going in.
The equation
r² = d² + (a / 2)²
r is the radius, d the distance from centre to chord, a the chord's length. The perpendicular, the radius and the half-chord form a right triangle; the equation is that triangle's Pythagoras. When d = 0 the chord is a diameter and a takes its largest value.
Same concept, elsewhere
Memorising one example gets you nowhere. You need to spot the concept wherever it turns up.
A tunnel's clear width at a given height — whether a lorry fits is decided by this calculation
The dipstick of a horizontal fuel tank: its marks are not evenly spaced, because width doesn't change linearly with depth
The phases of the Moon — the line dividing the lit part widens and narrows just like a chord
The width of a plank cut from a round log: the further from the centre you cut, the narrower it is
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