# Thales' Theorem Thales' Theorem is a mathematical concept used to determine unknown distances or heights. It relies on similar triangles. Discovered by ancient Greek philosopher and mathematician Thales (624-546 BCE), the theorem states: # Conditions 1. Two triangles: one large, one small. 2. Similar triangles (equal corresponding angles). 3. The smaller triangle forms from an altitude or line through a point on one side of the larger triangle. # Principle 1. Equal corresponding angles imply proportional sides. 2. Unknown distances/heights are determined using corresponding side ratios. # Applications 1. Calculating building or mountain heights. 2. Determining ship distances from shore. 3. Measuring tree heights. 4. Calculating civil engineering distances. # Implementation Steps 1. Draw similar triangles. 2. Identify corresponding angles. 3. Calculate corresponding side ratios. 4. Apply ratios to determine unknown distances. # Examples 1. Small triangle height: 3m, large triangle corresponding length: 9m; ratio 1:3. 2. Large triangle side: 15m; small triangle corresponding side: 5m.