The text and diagram in your image illustrate the use of similar triangles to determine the height of objects based on the length of their shadows. This method, attributed to Thales of Miletus, is used to find the height of tall structures by comparing their shadow with that of a known smaller object. Let's summarize the key points:

Summary:

• Thales' Method: Thales of Miletus used geometry and proportionality between the height of objects and the lengths of their shadows to determine unknown heights. This works by assuming the angles of sunlight hitting the objects are the same, creating similar triangles.

• Example Provided: The diagram shows two triangles: one representing a stick and its shadow, and the other representing a larger object (such as a pyramid) and its shadow.

  • The smaller triangle has a height of 1.5 meters and a shadow of 3 meters.
  • The larger triangle has a shadow of 4 meters, but its height is unknown.

• Proportions: Using the idea of similar triangles, we can set up a proportion between the corresponding sides of the triangles. This gives the equation: $\frac{\text{Height of small object}}{\text{Length of small shadow}} = \frac{\text{Height of large object}}{\text{Length of large shadow}}$ Substituting the known values: $\frac{1.5}{3} = \frac{\text{Height of large object}}{4}$

  Solving this equation will give