Introduction - Triangle Similarity

Relevance of the Topic

"In Mathematics, beauty lies in true simplicity. A triangle and a circle summarize all of geometry, with the addition of the number pi."

• Theodore von Karman

Triangle similarity is one of the main analytical tools that open the doors to the vast universe of geometry. It is a central concept that not only facilitates solving complex geometric problems but also applies in various practical situations, from technical drawing to maps and terrain graphs. Therefore, mastering triangle similarity is a critical step in the mathematical journey.

Contextualization

Triangles are two-dimensional polygons that are found abundantly in the real world, in the form of roofs, traffic signs, and even in electronic components. The study of triangle similarity follows the understanding of the basic elements of triangles - sides and angles. This is a natural step in the exploration of geometry, as it coordinates several elements into a single structure, allowing the application of mathematical principles in a more complex yet comprehensive way.

The concepts of triangle similarity were introduced in the 7th grade of Elementary School. Now, in the 9th grade, we are ready to explore these concepts in more depth, especially in situations where parallelism is a significant factor. Triangle similarity is also a fundamental basis for the introduction of trigonometric theorems in high school, making this scenario a vital preparation for future mathematical challenges.

Theoretical Development - Triangle Similarity

Components

• Similarity Ratio: The similarity ratio is a measure used to determine how large or small the image of an object is compared to the real object. It is defined as the ratio between the length of a pair of corresponding sides of the image triangle and the original triangle. For example, if the similarity ratio is 2:1, this means that the image triangle is twice the size of the original triangle.

• AAA Criterion (angle-angle-angle): This triangle similarity criterion states that if two triangles have all three corresponding angles congruent, then the triangles are similar. More specifically, if angles A and A' are congruent, angles B and B' are congruent, and angles C and C' are congruent, then triangles ABC and A'B'C' are similar.

• LAL Criterion (side-angle-side): This triangle similarity criterion states that if two triangles have a pair of corresponding sides proportional and the included angle between the sides congruent, then the triangles are similar. Therefore, if AB/A'B' = BC/B'C' and angle A is congruent to angle A', then triangles ABC and A'B'C' are similar.

Key Terms

• Triangle Similarity: Two triangles are similar if they have the same angles, meaning the internal angles of one triangle exactly correspond to the angles of the other. Additionally, the lengths of the corresponding sides of the triangles are proportional, meaning they are in the same ratio.

• Proportionality: It is a relationship of equivalence between two quantities. In the case of similar triangles, it implies that the lengths of the corresponding sides of the triangles are in the same proportion.

• Correspondence: The condition of correspondence is established when each element in one triangle has a corresponding element in the other. In triangle similarity, correspondence occurs between corresponding angles and sides.

Examples and Cases

• Example 1 - Applying the AAA Criterion: Consider two triangles with angle measures of 30°, 60°, and 90°. If the measure of each angle in one of the triangles is doubled, the new triangle formed is similar to the original triangle due to the AAA criterion.

• Example 2 - Applying the LAL Criterion: Consider two triangles with proportional sides BC = 2B'C' and AC = 2A'C' and congruent angles A = A'. Triangles ABC and A'B'C' are similar by the LAL criterion, as there is a proportionality relationship of the corresponding sides and the included angle between these sides is congruent.

• Case - Drawing a Triangle Similar in Larger or Smaller Scale: A practical application of triangle similarity is drawing triangles in a larger or smaller scale. This process depends on understanding and correctly applying the similarity ratio, which determines how much the drawn triangle should be expanded or reduced in relation to the original triangle.

Detailed Summary

Key Points

• Triangle Similarity: Two triangles are similar if their internal angles are congruent and the lengths of the corresponding sides are proportional. This is the fundamental principle you need to remember.

• Similarity Ratio: The similarity ratio is the relationship between the lengths of the corresponding sides of similar triangles. It is key to understanding the concept of similarity.

• AAA Criterion: If the angles of two triangles are congruent, then the triangles are similar, regardless of the side lengths. This is a useful criterion, easy to verify, and of great applicability.

• LAL Criterion: If one side of a triangle is proportional to a corresponding side of another triangle and the included angle between these sides is congruent, then the triangles are similar. This criterion combines the notion of proportionality with angle correspondence.

• Proportionality: It is essential to understand the idea that in similar triangles, the lengths of the corresponding sides are in the same proportion, which is expressed by a similarity ratio.

Conclusions

• Triangle similarity is a fundamental mathematical concept that applies to many geometric problems. It allows the understanding and manipulation of triangles on different scales.

• There are criteria (AAA and LAL) that allow us to identify if two triangles are similar. The AAA criterion is based solely on angle congruence, while the LAL criterion combines side proportionality with a congruent angle.

• The similarity ratio is a quantitative measure that determines the proportion of the lengths of the corresponding sides of similar triangles.

Suggested Exercises

1. Exercise 1: Consider two triangles with angle measures of 30°, 50°, and 100°. Using the AAA criterion, determine if the two triangles are similar.

2. Exercise 2: Let ABC be a triangle with angle measures of 30°, 60°, and 90°. If each side of the original triangle is doubled, determine if the resulting triangle is similar to ABC.

3. Exercise 3: In triangle ABC, angles A, B, and C measure 45°, 60°, and 75°, respectively. Determine the angles of a triangle similar to ABC, whose angles are twice the corresponding angles of ABC. Solve using the AAA criterion.