## Basic Proportionality Theorem & Similar TrianglesGeometry is the branch of mathematics that deals with the properties and relationships of shapes and figures; it is a concept that has fascinated thinkers and learners for centuries. One of the basic principles of Geometry is the concept of " " (BPT) is the core of this concept; it is a fundamental result that shows the relationship between line segments in a triangle. This theorem is also known as "Basic Proportionality Theorem," named after a famous Greek Mathematician, Thales Theorem. So, in this article, we will get to know the concept of the Thales explore its significance, proof, and applications, and conclude this article with a conclusion.Basic Proportionality Theorem,## Similar TrianglesBefore we get to learn about the Basic Proportionality Theorem or Thales Theorem, we have to get a basic understanding of a similar triangle because this theorem is based upon the concept of similar triangles; so, we can say that two triangles are similar if they follow two properties, which are discussed below: - If all the sides of a triangle are in proportion to each other.
- If all the angles which are correspond to the other triangle are equal.
Thus, we can say that two triangles, XYZ and triangle EFG, are similar when XY / EF = YZ / FG = XZ / EG Angle X = Angle E, Angle Y = Angle F, Angle Z = Angle G ## Understanding Proportionality in GeometryBefore we understand the Basic Proportionality Theorem, it is important to know the concept of proportionality in geometry. In geometry, proportions refer to the equality of ratios; a proportion is typically expressed as two equal ratios. When we say a, b, c, and d are in proportion, then we can write this as:
Here, 'a and b' are one pair of quantities, and 'c and d' are another pair; in the geometric context, these quantities are usually lengths or segments within a shape or figure. Proportions are used to show relationships between these lengths and are important in solving various geometric problems. Proportionality in geometry comes into existence from the concept of similar figures; two figures are considered similar if they have the same shape but may differ in size. When two figures are similar, their corresponding sides are proportional; in other words, the ratios of the lengths of corresponding sides are equal. Now that we understand proportionality in geometry let's explore the Basic Proportionality Theorem. ## The Basic Proportionality TheoremThe Basic Proportionality Theorem, usually referred to as Thales's Theorem, is a fundamental result in geometry that deals with the proportionality of line segments within a triangle; it states that " This theorem provides a powerful tool for analysing and solving problems involving triangles and parallel lines; to illustrate this theorem, we can consider a triangle ABC, with a line DE parallel to side BC, intersecting sides AB and AC at points D and E, respectively. In this concept, the Basic Proportionality Theorem asserts that:
In other words, the ratio of the length of segment AD to segment DB is equal to the ratio of segment AE to segment EC; this proportionality will remain consistent as long as the line DE is parallel to side BC. ## Proof of the Basic Proportionality TheoremIn this paragraph, we will try to prove the basic proportionality theorem; to do this, we first have to consider a triangle ABC shown in the image. In this triangle line,
So, the Area of the triangle ALM = 1 / 2 x AL x MX The Area of the triangle LBM = 1 / 2 x LB x MX The Area of the triangle ALM = 1 / 2 x AM x LY The Area of the triangle LMC = 1 / 2 x MC x LY So, the ratio of the areas of triangle ALM and triangle LMB: Area of the triangle ALM/Area of the triangle LBM = (1/2 x AL x MX) / (1/2 x LB x MX) Area of the triangle ALM / Area of the triangle LBM = AL / LB … (1) So, the ratio of the areas of triangle ALM and triangle LMC: Area of the triangle ALM/Area of the triangle LBC = (1/2 x AM x LY) / (1/2 x MC x LY) Area of the triangle ALM / Area of the triangle LBC = AM / MC … (2) As per the properties of triangles, So, this implies that the Area of the triangle LBM is equal to the Area of the triangle LMC, i.e., the Area of the triangle LBM = area of the triangle LMC … (3) So, from equation (1), (2), and (3) we conclude that: AL / LB = AM / MC Which we have to prove. ## Convers of the Basic Proportionality TheoremIn this paragraph, we will prove the converse of Thales's Theorem or Basic Proportionality Theorem, which states that if we draw a line that intersects the sides of the triangle and divide them proportionally, then the line will be parallel to the third side of the triangle. To prove this, first, consider a triangle ABC with E and D two points on the side of the triangle such that E lies on the AC side and D lies on the side AB, the line DE proportionally divides the sides of the triangle such that: AD / BD = AE / CE … (1)
So, from the Basic Proportionality Theorem, we can write: AD / BD = AE` / CE` … (2) From Equation (1) and (2), we can say that: AE / CE = AE` / CE` Adding one (1) both sides of the Equation, we get AE / CE + 1 = AE` / CE` + 1 (AE + CE) / CE = (AE` + CE`) / CE` AC / CE = AC / CE` CE = CE` This cannot be possible until points E and E` are the same, so we can conclude that points E and E` are the same point. This implies that DE parallels the BC, i.e., DE || BC, which we have to prove. ## Applications of the Basic Proportionality TheoremThe Basic Proportionality Theorem has a wide range of applications in geometry, providing a valuable tool for solving various problems involving triangles and parallel lines. Some common applications are discussed below: **Parallel Lines and Transversals:**The theorem is usually used to prove that some lines are parallel to each other by showing that they divide the sides of a triangle proportionally; for example, if you have a transversal line intersecting two parallel lines, you can use the theorem to demonstrate that the segments formed on the transversal line are proportional.**Similar Triangles:**The theorem is closely related to the concept of similar triangles, so it is used to prove that two triangles are similar by showing the proportionality of corresponding sides and other factors.**Geometric Constructions:**The theorem can be used in geometric constructions to divide line segments into proportional parts; this is particularly useful in creating geometric figures with specific proportions (also looks good).**Calculating Unknown Lengths:**When you know certain proportions within a triangle and have the length of one side, you can use the Basic Proportionality Theorem to calculate the lengths of other sides or segments within the triangle.**Coordinate Geometry:**The theorem can be applied in coordinate geometry to determine coordinates or equations of lines and points in the Cartesian plane.**Trigonometry:**In trigonometry, the Basic Proportionality Theorem is a fundamental concept when dealing with triangles and their side lengths and angles.
## Some Frequently Asked Questions
;" therefore, this theorem is usually referred to as "Thales Theorem" in his honour. Thales, who lived around 624-548 BC, made seven important contributions to geometry, and he is considered one of the earliest mathematicians in history.Thales
- Corresponding angles of both triangles are equal: This means that the angles at the vertices of the triangles must have the same measures; corresponding angles occupy the same relative positions in the two triangles.
- Corresponding sides of both triangles are in proportion to each other: This condition implies that the ratios of the lengths of corresponding sides in the two triangles are equal. In other words, if you take any pair of sides from both triangles, the ratios of their lengths must be the same.
## ConclusionThe is basic and very important in geometry; its simplicity makes it an invaluable tool for understanding the proportionality of line segments within triangles and other geometric figures. By providing a foundation for proving the equality of ratios, this theorem plays a crucial role in solving a wide range of geometric problems, from demonstrating the parallelism of lines to proving the similarity of triangles; its applications lie far beyond the classroom, finding usefulness in various fields, including architecture, engineering, and surveying. As a fundamental principle in geometry, the Basic Proportionality Theorem reminds us of the power and beauty of mathematical concepts in showing the mysteries of the world around us.Thales's Theorem,Next TopicCayley-Hamilton Theorem |