Geometric similarity in triangles

Geometric similarity in triangles . Two triangles are similar if there is a similarity or similarity relationship between them.

Summary

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• 1 Definitions
• 2 equation
  • 1 Corollaries
• 3 Fundamental theorem of the similarity of triangles
  • 1 Cases
  • 2 First case
  • 3 Second case
  • 4 Third case
• 4 See also
• 5 Sources

Definitions

A similarity is the composition of a matter (a rotation and a possible reflection or axial symmetry) with a homothecia. In rotation you can change the size and orientation of a figure but its shape is not altered.

Therefore, two triangles are similar if they have a similar shape.

In the case of the triangle, the shape only depends on its angles (not so in the case of a rectangle, for example, where one of its angles is right but whose shape can be more or less elongated, that is, it depends on the base quotient / height).

The definition can be simplified in this way: two triangles are similar if their angles are equal two to two.

In the figure, the corresponding angles are A = A ‘, B = B’ and C = C ‘. To denote that two triangles ABC and DEF are similar, we write ABC ~ DEF, where the order indicates the correspondence between the angles: A, B and C correspond to D, E and F, respectively.

A similarity has the property (which characterizes it) of multiplying all lengths by the same factor. Therefore, the image length / origin length ratios are all the same, which gives a second characterization of similar triangles:

Two triangles are similar if the ratios of the corresponding sides are congruent.

Equation

These two equivalent properties are gathered in the following equation:

Demonstrative equation of equivalent properties.

Corollaries

• All equilateral triangles are similar.
• If two triangles have two equal angles, the third ones are also equal.

A similarity is the composition of an isometry (a rotation and a possible reflection or axial symmetry) with a homothecia. In rotation you can change the size and orientation of a figure but its shape is not altered. Therefore, two triangles are similar if they have a similar shape. In the case of the triangle, the shape only depends on its angles (not so in the case of a rectangle, for example, where one of its angles is right but whose shape can be more or less elongated, that is, it depends on the base quotient / height). The definition can be simplified in this way: two triangles are similar if their angles are equal two to two.

In the figure, the corresponding angles are A = A ‘, B = B’ and C = C ‘. To denote that two triangles ABC and DEF are similar, we write ABC ~ DEF, where the order indicates the correspondence between the angles: A, B and C correspond to D, E and F, respectively. A similarity has the property (which characterizes it) of multiplying all lengths by the same factor. Therefore, the image length / origin length ratios are all the same, which gives a second characterization of similar triangles.

Two triangles are alike if the ratios of the corresponding sides are congruent Reflective, reflective or identical property Every triangle is similar to itself. Identical or symmetric property If one triangle is similar to another, it is similar to the first. Transitive property If a triangle is similar to another, and this triangle in turn is similar to a third, the first is similar to the third. These three properties imply that the similarity relationship between two triangles is an equivalence relationship.

Fundamental theorem of the similarity of triangles

All the parallels to one side of a triangle that does not pass through the opposite vertex, determine with the lines to which the other two sides belong, a triangle similar to the one given.

H)

ABC; r || AC
r cuts AB in L
r cuts BC in M

T)

1. D)

Cases

3 cases may be presented:

First case

r cuts sides AB and BC at points inside them.

We will make a first consideration, referring to the angles, and we will call it (1):

by reflex character

for being corresponding between r || BC, secant AB

for being corresponding between r || BC, secant AC

On the other hand, by virtue of the corollary of Thales’ Theorem we have:

If by M a parallel is drawn to the side AB, it intersects the side AC at a point N, and again by the corollary of Thales’ Theorem we have:

But since AN = LM, being opposite sides of the ALMN parallelogram, replacing in

is obtained:

From and we get the consideration that we will call (2):

After (1) and (2), it results:

by definition of similarity.

Second case

r cuts the lines on the sides AB and BC at points outside them, on the rays of origin B that contain them.

We consider BLM as if it were the given triangle, and BAC the new triangle, and for the case I of the proof, it is:

by symmetrical character.

Third case

r cuts the lines on the sides AB and BC at points that belong to the opposite rays to those that support the sides.

On the ray of origin B that contains point A, BN ​​= BL is constructed and at the end N of the constructed segment, a parallel to AC (s) that cuts the line of BC by O. Then there are cases I, similarity which we will call \ otimes.

Taking into account the triangles BNO and BLM, it is observed:

• BN = BM by construction
  * α = α ‘for being opposed by the vertex.
  * β = β ‘for being internal alternates between r || s, secant MN

And being BNO = BLM is BNO ~ BLM \ oplus by the first corollary of the definition.

From \ otimes and \ oplus, and for a transitive nature:

BAC ~ BLM \ Rightarrow BLM ~ BAC