Triangle Proportionality Theorem
The Triangle Proportionality Theorem is a fundamental concept in geometry that describes the relationship between the sides of a triangle and a line drawn parallel to one of its sides. According to the theorem, if a line is drawn parallel to one side of a triangle and intersects the other two sides, it divides those sides into segments that are proportional. For instance, if one side of the triangle is divided into segments of 1/3 and 2/3, the other side will also be divided in the same ratio. This theorem not only helps in finding missing lengths in geometric figures but also establishes that if two sides of a triangle are divided proportionally, then the intersecting line must be parallel to the third side. The converse is also valid; if the segments are not proportional, then the line is not parallel. This theorem plays a crucial role in various mathematical applications, serving as a key principle for understanding proportional relationships in triangles. Overall, the Triangle Proportionality Theorem is an essential tool for students and anyone studying the properties of triangles and their geometric relationships.
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Subject Terms
Triangle Proportionality Theorem
Proportionality is a mathematical relation. Two variables are proportional if a change in one always results in or is otherwise accompanied by a change in the other, in which the changes are related by a constant multiplier. The constant is called the proportionality constant, constant of variation, or coefficient of proportionality. An obvious and familiar example of proportionality is a map drawn to scale. By definition, the distance between two points on the map (say, your home and the movie theater) is proportional to the distance between those two places in real life. Further, this is true both of the straight-line distance between them and the length of the route taken over streets and other means of travel. A more scientific example of proportionality is the force of gravity acting on an object, which is directly proportional to the object's mass, with gravitational acceleration acting as the proportionality constant.
If one variable is the product of the other, they are directly proportional, as when x = 2y (2 being proportionality constant). If the product of two variables is equal to a constant, they are inversely proportional to one another. If xy = 2, for instance, then y = 2/x and x = 2/y. The symbol ∝ is used to show proportionality, as in y ∝ x.
Inverse proportionality is also easily illustrated. The time taken to drive from point A to point B is inversely proportional to the speed at which one drives; as speed increases, the time of the journey decreases.
Overview
Though his name is not as familiar as that of Pythagoras or Euclid, Eudoxus of Cnidus was one of the greatest Greek mathematicians, and contributed significant work on proportionality in arithmetic and geometry that, when rediscovered in sixteenth century Europe, remained the most advanced work on the topic for another century. His work addressed proportionalities not only in two-dimensional space but also in prisms, pyramids, spheres, and cones, as well as developed ways to address continuous geometric objects such as lines and angles, and helped to popularize axiomatic deduction.
The triangle proportionality theorem expresses a relationship between elements of a triangle. It states that a line drawn parallel to any of the sides of the triangle and intersecting the other two sides will divide them proportionally. In other words, though the segments that the intersecting line crosses may be of different lengths, if side AB is divided into segments of 1/3 and 2/3 of its length, then so will side AC. Or, generally, where line DE is parallel to line BC in triangle ABC, and intersects line AB and line AC, AD/DB = AE/EC.The triangle proportionality theorem is often taught to students by having them use it to find the missing lengths of line segments in a triangle diagram.
The converse is also true. If a line intersecting two sides of a triangle divides each side proportionally, then it is parallel to the third side. Of course, the contrapositive is also true. If the sides are not divided proportionally, the line is not parallel to the third side.
Bibliography
Aigner, Martin, and Gunter M. Ziegler. Proofs from the Book. New York: Springer, 2014.
Grunbaum, Branko, and G. C. Shephard. Tilings and Patterns. New York: Dover, 2015.
Hanna, Gila. Explanation and Proof in Mathematics. New York: Springer, 2014.
Kahn, David. Attacking Trigonometry Problems. New York: Dover, 2015.
Larson, Ron. Algebra and Trigonometry. Boston: Cengage, 2015.
Millman, Richard, Peter Shiue, and Eric Brendan Kahn. Problems and Proofs in Numbers and Algebra. New York: Springer, 2015.
Perrin, Daniel. Algebraic Geometry: An Introduction. New York: Springer, 2008.