Intercept theorems

How tall is the tree outside your window, really? You cannot reach up to measure it and you do not want to climb it. Yet a folding rule, a straight stick and a bit of sunshine are enough to get the height surprisingly right. Behind this sit the intercept theorems: two rules about ratios that apply whenever parallel lines cut two rays.

Picture a point $S$ with two rays coming out of it, like the two arms of an open pair of scissors. Across them you lay two lines that are parallel to each other. The nearer one meets the first ray at $A$ and the second ray at $B$, the farther one meets those same rays at $C$ and $D$. So on the first ray you get $S$, $A$ and $C$ in that order, on the second ray $S$, $B$ and $D$. This creates two triangles sharing the same tip $S$, a small one and a large one. Because the cutting lines are parallel, the large triangle is nothing but an enlarged copy of the small one, and that is exactly why all the lengths are linked by fixed ratios.

The first intercept theorem

The first intercept theorem compares the segments on the two rays. One thing is crucial here: every length is measured from the vertex $S$, not from one parallel to the next. If $SA = 4$ cm and $SC = 6$ cm, then $C$ lies 1.5 times as far from $S$ as $A$ does. On the second ray exactly the same factor applies: $SB = 6$ cm becomes $SD = 9$ cm. Once you know three of the four lengths, you can always find the fourth.

The second intercept theorem

The second intercept theorem brings in the parallels themselves. The segment $AB$ on the near parallel and the segment $CD$ on the far one are in the same ratio as the segments on either of the two rays. With $SA = 4$ cm, $SC = 6$ cm and $AB = 5$ cm it follows that $CD$ is 1.5 times 5 cm, so 7.5 cm. Remember the difference like this: the first theorem stays on the rays, the second links a ray with the crosswise pieces.

Measuring the tree without climbing

Sunlight reaches us practically parallel. Stand a stick upright next to the tree and the stick with its shadow and the tree with its shadow form the same figure in two sizes, with the sun taking over the role of the rays. A 1.5 m stick casts a 1.2 m shadow, so height to shadow is 1.5 to 1.2, which is 1.25 times. If the tree casts an 8 m shadow at that same moment, it is 8 m times 1.25, so 10 m tall. The only thing that matters is measuring both shadows at once, because the sun moves on and every shadow with it.

The same figure also works the other way round, when it is a distance you are after. Hold a pencil in front of your eye so that it exactly covers a tower, and your eye is the vertex $S$, the two lines of sight over the ends of the pencil are the rays, and the pencil and the tower are the two parallels. The pencil tells you the ratio of distance to length, and the tower follows the very same ratio.