Intercept Theorem Explained Simply
The intercept theorems describe what happens when two lines start from a common point and are cut by two parallel lines. Two triangles appear that share the same shape but differ in size — and because the shape is identical, all lengths stand in the same ratio.
Why it works
Parallel lines meet both rays at exactly the same angle, and the angle at the apex belongs to both triangles anyway. All three angles therefore agree, which makes the triangles similar: one is nothing but an enlarged copy of the other, with every length multiplied by the same factor.
Two rays from the apex Z are cut by two parallel segments. The ratios of the pieces are equal.
Scale factor k = ZB / ZA = 12 / 4 = 3
BD = k · AC = 9 cm
Check: 4 : 12 = 3 : 9 → both sides give 1/3 ✓
Measuring a tree by its shadow
A stick 1.50 m tall casts a shadow of 2.00 m. At the same moment a tree casts a shadow of 14.00 m. Over that distance the sun's rays are effectively parallel, so stick and tree form two similar triangles. The tree is therefore 14.00 · 1.50 / 2.00 = 10.50 m tall. Thales is said to have measured the Great Pyramid this way.
| What is compared | Ratio | |
|---|---|---|
| First theorem | segments on the same ray | ZA : ZB = ZC : ZD |
| First theorem (variant) | pieces between the parallels | ZA : AB = ZC : CD |
| Second theorem | parallels against ray segments | AC : BD = ZA : ZB |
Mixing up the segments. ZB is measured from the apex, AB is only
the piece in between.
Not checking that the lines are parallel.
Setting the ratio up crosswise. Both fractions must be built the same
way round.
A person 1.80 m tall stands 3.00 m from a street lamp and casts a 2.00 m shadow. How high is the lamp?
Show solution
The apex is the tip of the shadow. Distance to the person 2.00 m, to the lamp post
5.00 m.
h / 1.80 = 5.00 / 2.00 = 2.5 → h = 4.50 m
Pythagorean theorem · Sine, cosine and tangent · Area and volume formulas