Understanding Proportionality and Similarity in Geometry
Two quantities, X and Y, are proportional if their ratio is constant: y / x = m (or equivalently, y = mx). In this case, the number m is called the constant of proportionality. The graph of the function relating two proportional quantities is always a straight line through the origin, represented by the equation y = mx.
Two figures are similar if they have the same shape, though different in size. Two conditions must be met for figures to be considered similar:
- The corresponding segments are proportional; that is, the length of each segment in one figure is obtained by multiplying the corresponding length in the other figure by a fixed rate (e.g., double, triple). This constant is the ratio of similarity.
- The corresponding angles must be equal.
Thales’ Theorem
Thales’ Theorem states that when two secant lines are intersected by a series of parallel lines, the segments formed on one line are proportional to the corresponding segments on the other line.
For example, in a geometric configuration where OA, OB, and OC are segments on one secant line, and OA’, OB’, and OC’ are corresponding segments on another secant line, intersected by parallel lines, the following proportions hold:
OA / OA’ = OB / OB’ = OC / OC’ = AB / A’B’ = AC / A’C’ = BC / B’C’ = constant
From OA / OA’ = OB / OB’, we can deduce that OA * OB’ = OB * OA’ (product of means equals the product of extremes), and consequently, OA / OB = OA’ / OB’.
This leads to another way of stating Thales’ theorem:
Thales’ Theorem (Second Statement): When two secant lines are intersected by a series of parallel lines, the ratio between any two segments on one line is equal to the ratio of the corresponding segments on the other line.
Any line parallel to one side of a triangle that intersects the other two sides creates proportional segments on those sides. In triangle ABC, if a line parallel to BC passes through points D and E, the segments formed are proportional because their ratios are equal.
Similarity of Triangles
Two triangles are similar if they have equal angles and proportional sides. If triangles ABC and A’B’C’ are similar, then angles A = A’, B = B’, and C = C’, and the proportions A’B’ / AB = B’C’ / BC = C’A’ / AC = r hold, where r is the similarity ratio.
While two figures are similar if their corresponding segments are proportional and their corresponding angles are equal, for triangles, meeting just one of these conditions is sufficient (as the other condition is automatically satisfied). This is not the case for other polygons.
Criteria for Triangle Similarity
- LLL Criterion: If two triangles have proportional sides, they are similar.
- AAA Criterion: If two triangles have equal angles, they are similar.
These concepts of triangle similarity are fundamental to defining trigonometric ratios.