Thales' Theorem
Senior 2 · Mathematics
Thales' Theorem
Lesson 4 of 4
22 min
How tall is the tallest eucalyptus tree at your school? How wide is the Nyabarongo river at a point where you cannot cross? You cannot put a tape measure on either, but a stick, a tape measure and Thales' theorem will give you both answers.
On a sunny day, the sun's rays are parallel. Stand a stick upright so that the tip of its shadow falls exactly at the tip of the tree's shadow, at the point P. The stick and the tree are both vertical, so they are parallel, and Thales' theorem applies to the triangle formed by the tree, the ground and the sun ray through P.
Worked example 1
A stick 1.5 m tall has a shadow 2 m long. The tree's shadow, measured from P, is 18 m. Find the height h of the tree.
stick ÷ tree = stick's shadow ÷ tree's shadow, so 1.5/h = 218. Then 2h = 27 and h = 13.5 m.
A student stands on the near bank at B, exactly opposite a tree A on the far bank. She walks straight back from the river 10 m to E. From B she walks 12 m along the bank to C. From E she walks parallel to the bank until she reaches D, where C lies on her line of sight to the tree. ED is 15 m.
Worked example 2
BC ∥ ED, so by Thales' theorem in triangle AED: AB/AE = BC/ED. Write the width as w: w/(w + 10) = 1215.
Cross-multiply: 15w = 12w + 120, so 3w = 120 and w = 40 m. The river is 40 m wide at that point.
To divide a segment AB into 3 equal parts with only a ruler and compasses:
This works because the parallel lines cut both lines in the same ratio (Thales' theorem), so equal steps on the ray give equal steps on AB.
Tip
Exam technique. In a word problem, first draw the figure and mark the parallel lines. Then write one ratio equation with the unknown in it, labelling each length with its meaning. Check that the answer is sensible: a tree taller than its stick-to-shadow ratio suggests, or a river width smaller than zero, means a ratio was upside down.
At the same moment, a stick 2 m tall casts a shadow 1.6 m long and a building casts a shadow 20 m long. How tall is the building?
Solution: h/2 = 201.6, so h = 2 × 20 ÷ 1.6 = 25 m.
In the river set-up, BE = 8 m, BC = 9 m and ED = 12 m. Find the width w = AB.
Solution: w/(w + 8) = 912. So 12w = 9w + 72, 3w = 72 and w = 24 m.
Why must the stick be vertical for the shadow method to work?
Solution: The tree is vertical. Thales' theorem needs the stick to be parallel to the tree; a leaning stick would not be parallel, and the ratios would no longer be equal.
Remember
Sources and credits
The Rwandan worked examples, the practice items and the exercises were written by Ganzaa.
