The idea of similarity is an important one in geometry. Similar figures are figures that are the exact same shape, but are different sizes. In different words, two figures are similar if one figure can be obtained from the other by uniformly scaling (enlarging or reducing), possibly with additional translation, rotation and reflection. For example, all circles are similar to each other, all squares are similar to each other, and all equilateral triangles are similar to each other. On the other hand, ellipses are not all similar to each other, rectangles are not all similar to each other, and isosceles triangles are not all similar to each other.
Most mathematicians consider congruent shapes to also be similar (going beyond similarity to be identical in size). Yet it must be kept in mind that, though congruent shapes are automatically similar, similar shapes are not automatically congruent. The most common way to use similar figures is knowing the size of a small figure can help you estimate the size of a larger, unknown measurement. Similar triangles provide the basis for many proofs in Euclidean geometry. We will discuss similar polygons in later sections but before we move on to that we need to understand another key concept in mathematics, the idea of ratio and proportion.
Proportion
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.
Proportion Theorems
- In any proportion the product of the extremes is equal to the product of the means.
- The mean proportional between two quantities is equal to the square root of their product.
- If the two antecedents of a proportion are equal, the two consequents are equal.
- If the product of two quantities is equal to the product of two others, either two may be made the extremes of a proportion in which the other two are made the means.
- One side of a triangle is to either of its segments cut off by a line parallel to the base as the third side is to its corresponding segment.
- If a line divides two sides of a triangle proportionally from the vertex, it is parallel to the third side.
- The bisector of an angle of a triangle divides the opposite side into segments which are proportional to the adjacent sides.
- The bisector of an exterior angle of a triangle divides the opposite side externally into segments which are proportional to the adjacent sides.
- For every angle in one of the figures there must be an equal angle in the other.
- The corresponding sides must be proportional.
- Two mutually equiangular triangles are similar.
- Two triangles are similar if two angles of the one are equal respectively to two angles of the other.
- Two right triangles are similar if an acute angle of the one is equal to an acute angle of the others
- If two triangles have an angle of the one equal to an angle of the other, and the including sides proportional, they are similar.
- If two triangles have their sides respectively proportional they are similar.
- Two triangles which have their sides respectively parallel, or respectively perpendicular, are similar.
- The perimeters of two similar polygons have the same ratio as any two corresponding sides.
- If two polygons are similar, they can be separated into the same number of triangles, similar each to each, and similarly placed.
- If two polygons are composed of the same number of triangles, similar each to each, and similarly placed, the polygons are similar.
- If in a right triangle a perpendicular is drawn from the vertex of the right angle to the hypotenuse:
- The triangles thus formed are similar to the given triangle, and are similar to each other.
- The perpendicular is the mean proportional between the segments of the hypotenuse.
- Each of the other sides is the mean proportional between the hypotenuse and the segment of the hypotenuse adjacent to that side.
- The squares on the two sides of a right triangle are proportional to the segments of the hypotenuse adjacent to those sides.
- The square on the hypotenuse and the square on either side of a right triangle are proportional to the hypotenuse and the segment of the hypotenuse adjacent to that side.
- The perpendicular from any point on a circle to a diameter is the mean proportional between the segments of the diameter.
- If a perpendicular is drawn from any point on a circle to a diameter, the chord from that point to either extremity of the diameter is the mean proportional between the diameter and the segment adjacent to that chord.
- If two chords intersect within a circle, the product of the segments of the one is equal to the product of the segments of the other.
- If from a point outside a circle a secant and a tangent are drawn, the tangent is the mean proportional between the secant and its external segment.
- The square on the bisector of an angle of a triangle is equal to the product of the sides of this angle diminished by the product of the segments made by the bisector upon the third side of the triangle.
- In any triangle the product of two sides is equal to the product of the diameter of the circumscribed circle by the altitude upon the third side.
- At B erect a perpendicular BE equal to half of AB.
- From E as a center, with a radius equal to EB, describe a circle.
- Draw AE, meeting the circle at F and G.
- On AB take AC equal to AF.
- On BA produced take AC’ equal to AG.