Basic Proportionality Theorem & Similar Triangles

Geometry 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 "Proportionality," which is very important in understanding the properties of triangles and other geometric shapes. The "Basic Proportionality Theorem" (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 "Thales Theorem," named after a famous Greek Mathematician, Thales. So, in this article, we will get to know the concept of the Basic Proportionality Theorem, explore its significance, proof, and applications, and conclude this article with a conclusion.

Similar Triangles

Before 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:

1. If all the sides of a triangle are in proportion to each other.
2. 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 Geometry

Before 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:

a / b = c / d

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 Theorem

The 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 "if a line is parallel to one side of a triangle and also intersects the other two sides, then that line will divide those two sides proportionally."

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:

AD / DB = AE / EC

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 Theorem

In 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, LM is parallel to the line BC and the intersecting side of triangle ABC, L on the side AB, and M on the side AC.

To Prove: Per the basic proportional theorem, we must show that AL / LB = AM / MC.

Construction: Within the triangle ABC, join the vertex B to M and, similarly, join vertex C to L to form lines BM and CM and then draw a perpendicular MX to the side AL and drop a perpendicular LY to the side AM, as shown in the figure below.

Proof: The Area of a triangle is equal to half of the base and height of that triangle, i.e., 1 / 2 x base x height.

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, those triangles that are made on the same base and between two parallel lines all have equal Area.

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 Theorem

In 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)

To Prove: We have to prove that the line DE parallels the base of the triangle BC, i.e., DE || BC.

Construct: To show this, let us draw a line DE`, such that DE` is parallel to BC, i.e., DE` || BC. As shown in the image below:

Proof: As we have constructed, DE` is parallel to BC in the triangle ABC.

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 Theorem

The 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:

1. 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.
2. 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.
3. 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).
4. 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.
5. Coordinate Geometry: The theorem can be applied in coordinate geometry to determine coordinates or equations of lines and points in the Cartesian plane.
6. 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

Q: What do you understand by the Basic Proportionality Theorem?

Answer: The Basic Proportionality Theorem, usually written as BPT, is a fundamental geometric principle that deals with the proportionality of line segments within triangles; it states that if a line is drawn parallel to one side of a triangle, intersecting the other two sides at distinct points, then the line divides those two sides in the same ratio. In other words, the ratios of the lengths of corresponding line segments on the two sides remain constant as long as the line is parallel to the third side of the triangle; this theorem is very important in geometry and is particularly useful for proving parallelism and similarity of triangles.

Q: Who was behind the discovery of the Basic Proportionality theorem?

Answer: The Basic Proportionality Theorem, also known as Thales's Theorem, was introduced by the famous Greek mathematician "Thales;" 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.

Q: What is the corollary of the BPT (Basic Proportionality) theorem?

Answer: The corollary of the Basic Proportionality Theorem (BPT) states that if a line divides any two sides of a triangle in the same ratio, then that line is parallel to the third side of the triangle; in other words, when a line intersects two sides of a triangle in such a way that the ratio of the segments it forms on those sides is constant, it implies that the line is parallel to the remaining side of the triangle.

Q: What is another name for the Basic Proportionality Theorem?

Answer: Another name for the Basic Proportionality Theorem (BPT) is Thales's theorem. This name is usually used in honour of the ancient Greek mathematician Thales; it describes the proportionality of line segments within triangles when a line is drawn parallel to one side and intersects the other in distinct points. Thales's contributions to geometry have had a lasting impact, and his name is associated with several (in particular, seven) fundamental theorems.

Q: What are the conditions for two triangles to be similar?

Answer: For two triangles to be similar to each other, two conditions must be met:

1. 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.
2. 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.

Q: Is it necessary that when two triangles are similar, they have the same size?

Answer: Similar triangles don't need to have the same size; while they have the same shape, their sizes can vary. The key property of similar triangles is that the ratios of the lengths of their corresponding sides are equal. This means that we can scale one triangle up or down uniformly to obtain the other; in other words, similar triangles are proportional in size but not necessarily congruent, which would mean having both the same shape and the same size. Similar triangles are basic in geometry and trigonometry and are usually used to solve problems involving indirect measurements and proportionality.

Conclusion

The Basic Proportionality Theorem, also known as Thales's Theorem, 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.