✅ point 1
Proportion is the expression of equality between two equal ratios. Where ratio is defined as the comparison or simplified form of two quantities of the same kind. This relation indicates how many times one quantity is equal to the other; or in other words, ratio is a number which expresses one quantity as a fraction of the other. A ratio is an indication of the relative size of two magnitudes.
Symbols: A proportion is written in one of the following forms: a : b = c : d; a : b :: c : d; a/b = c/d. This proportion is read ” a is to b as c is to d “; or ” the ratio of a to b is equal to the ratio of c to d.”
Terms: In a proportion the four quantities compared are called the terms. The first and third terms are called the antecedents; the second and fourth terms, the consequents.
The first and fourth terms are called the extremes; the second and third terms, the means. Thus in the proportion a : b :: c : d, a and c are the antecedents, b and d the consequents, a and d the extremes, b and c the means.
Fourth Proportion: The fourth term of a proportion is called the fourth proportional to the terms taken in order. Thus in the proportion a:b = c: d, d is the fourth proportional to a, b, and c.
Continued Proportion: The quantities a, b, c, d, e are said to be in continued proportion, if a : b = b : c = c : d =.d : e. If three quantities are in continued proportion, the second is called the mean proportional between the other two, and the third is called the third proportional to the other two. Thus in the proportion a: b = b: c, b is the mean proportional between a and c, and c is the third proportional to a and b.
Theory of proportion – Proportion applied to similar figures
Some euclidean definitions:
- Similar rectilinear figures are such as have their angles severally equal and the sides about the equal angles proportional.
- Two figures are reciprocally related when the sides about corresponding angles are reciprocally proportional.
- A straight line is said to have been cut in extreme and mean ratio when, as the whole line is to the greater segment, so is the greater to the less.
- The height of any figure is the perpendicular drawn from the vertex to the base.