Solving Similar Triangles
Solving similar triangles involves understanding the relationship between the angles and side lengths of two triangles that share the same shape but may differ in size. In geometry, two triangles are considered similar if all their corresponding angles are equal, leading to proportionality in their side lengths. This means that the ratios of the lengths of corresponding sides in similar triangles are constant. The notation A ~ B signifies that triangles A and B are similar. The concept of similarity is crucial in various fields, including physics and engineering, as it allows for the determination of unknown measurements through proportional relationships. Historical context shows that the study of similarity dates back to Euclid, who emphasized its importance in geometry. Additionally, it's important to distinguish between similarity and congruence; while congruent triangles are identical in both shape and size, similar triangles share the same shape regardless of their dimensions. Understanding these principles provides a valuable tool for simplifying geometric problems and finding elegant solutions.
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Solving Similar Triangles
The study of geometry is not only the study of shapes, but also the study of the relationship between various shapes. It is essential to discover ways to simplify a particular problem with the ability to be able to filter out exactly what is needed. Recognizing similarity can help do just this. In geometry, two objects are similar if they have the same shape. One object may be a different size than the other, but it may not be a different shape in order for the two to be similar. This concept is expected knowledge of students in physics, engineering, and the sciences, because its simple statement is rather useful in finding unknown measurements, and a highly useful one at that.
It is thought that geometrical similarity first began to be used as a tool during the time of Euclid in the third century in his Book of Elements. Euclid described in detail the importance of the study of similarity and how one needs to perform several calculations in order to arrive at a simpler solution. Engineers and geometers have developed tools to look for properties of objects such as similarity so that a problem at hand may have a shorter and more elegant solution.
Overview
In two similar triangles, all of the three angles will be equivalent, which causes the ratios of the respective sides on each triangle to be identical. Angles thus describe what relative shape which is being formed, but not the actual scale factor, or size of the object. This means that each of the pairs of parallel side lengths on the two triangles will have the exact same value when taking the ratio between the two.
Mathematically, two geometrical objects A and B are represented as being similar with the notation A
B. Therefore, there exists a constant value one can multiply each side on object A to obtain each corresponding side on object B. In Figure 1, since all of the angles are equivalent, the two triangles are similar.
Since ABC
DEF then one can find sides d and f by solving the two equations:
Rotation and reflection are independent of whether triangles, or any geometrical objects are similar. Thus, if two triangles are similar, any rotation or reflection of the triangles is also similar, since the form of the objects are remained the same.
Congruence and similarity also have different meanings in geometry. Congruence between objects describes that the objects are exactly of the same shape and size, whereas similarity describes the object only being of the same shape, and not necessarily the same size. Therefore all objects that are congruent are also similar, but not all objects which are similar are congruent, since their side lengths may differ.
Bibliography
Alonso, Orlando B., and Joseph Malkevitch. "Classifying Triangles and Quadrilaterals." The Mathematics Teacher 106.7 (2013): 541.
McKellar, Danica. Girls Get Curves. New York: Penguin, 2012.
Posamentier, Alfred S., and Robert L. Bannister. Geometry, Its Elements and Structure. Mineola, NY: Dover, 2014.
Wolfe, Harold E. Introduction to Non-Euclidean Geometry. Mineola, NY: Dover, 2012.