Basic Proportionality Theorem (BPT) Proof & Examples

The basic Proportionality Theorem is one of the most important theorems used in geometry, which is related to the length of the sides of triangles. The theorem was introduced by the famous Greek mathematician Thales.

• The basic proportionality theorem is also known as Thales's theorem.
• According to Thales, the ratio of any two corresponding sides of any given triangle is always the same. 
• It is based on the concept of similar triangles.
• In this, the corresponding angles of both the triangles are equal.
• The theorem has been used in many real life situations.
• The fare that every individual pays for the train journey.

Key Terms: Triangle, Basic Proportionality Theorem, BPT, Side Splitter Theorem, Intercept Theorem, Similar triangles, Elementary Geometry, Base, Height, Proof of Basic Proportionality Theorem, Tangent

---

Statement of Basic Proportionality Theorem

[Click Here for Sample Questions]

Basic Proportionality Theorem states that when a line parallel to the third of the side of a triangle passes through the triangle, then the parallel line divides the triangle into two equal sides (i.e., in equal ratio).

• Apart from the Basic Proportionality Theorem and Thales Theorem, other names of BPT are the Side Splitter Theorem and the Intercept Theorem.
• It is one of the important theorems in Elementary Geometry when it comes to finding one of the lengths of a triangle. 
• This theorem is also equivalent to the theorem on ratios of similar triangles.
• In a similar triangle, the corresponding sides of both triangles are in proportion to each other.
• Similarly, the corresponding angles of both triangles are equal.
• As per the below figure, basic proportionality theorem implies that

AD/DB = AE/EC.

Basic Proportionality Theorem

Also Read:

---

Proof of Basic Proportionality Theorem

[Click Here for Sample Questions]

One of the simple ways to prove the basic proportionality theorem is as follows:

Statement: The line drawn, which is parallel to one side of a triangle and drawn by cutting the other two sides of a triangle, divides the other two sides in equal proportion. 

To Prove: In the given Triangle prove that: AD/DB = AE/EC

Given: In the given triangle draw EN perpendicular to AD and similarly, DM Perpendicular to AC. Also, join DC with BE.

Proof: As we know Area of a Triangle is 1/2 ×base× height

• Now in the given triangle ABE,

Area of Triangle ADE= 1/2 ×AE×DM ---------(1)

• Also,

Area if Triangle ADE=1/2×AD×EN ---------(2)

• Similarly,

Area of Triangle BDE=1/2×EN×BD---------(3)

Area of Triangle CED=1/2×EC×DM---------(4)

• Dividing (1) by (4)
• Area of Triangle ADE/Area of Triangle CED = (1/2 ×AE×DM)/ (1/2×EC×DM)
• we can see that DM cancels out along with ½.
• And we get Area of Triangle ADE/Area of Triangle CED

AE/EC---------(5)

• Similarly divide (2) by (3)
• Area of Triangle ADE/Area of Triangle BDE =(1/2×AD×EN)/(1/2×EN×BD)
• We get Area if Triangle ADE/Area of Triangle BDE

AD/BD---------(6)

• Since the base of Triangles CED and BDE are the same i.e., DE, and is also between the same parallel lines DE and BC hence,
• Area of Triangle CED=Area of Triangle BDE---------(7)
• Hence, from (5), (6), (7)

AE/EC=AD/DB

Hence proved.

Proof of Basic Proportionality Theorem

---

Converse of BPT

[Click Here for Sample Questions]

The converse of BPT are as follows:

Statement: According to the converse of basic proportionality theorem, "If a line segment is drawn to cut two sides of a triangle in equal proportion, then it is parallel to the third side".

Given: ABC is a triangle and the line DE cuts the sides AB and AC in equal proportion. AD/BD = AE/CE

Proof: DE is not parallel to BC. Hence, construct another line DF that is parallel to BC. By the Basic Proportionality theorem, we have: AD/BD = AF/CF.

• But it is given that AD/BD = AE/CE.

From the above two expressions the left-hand side of both the expressions are same hence we can conclude that right hand side will also be in equal proportion.

• Hence, AE/CE = AF/CF.
• Adding 1 on both sides of this statement.
• (AE/CE) + 1= (AF/CF) + 1
• (AE + CE)/CE = (AF + CF)/CF
• AC/CE = AC/CF
• CE = CF

From the above statements, the points E and F are the same points and are coincident. Hence the line DE is parallel to BC and hence it proves the converse of the basic proportionality theorem.

Converse of BPT 

---

Applications of Basic Proportionality Theorem

[Click Here for Sample Questions]

The basic Proportionality Theorem can also be used in real-life problems such as:

• BPT can be used to measure the length of the tree’s shadow and your shadow.
• It can also be used to construct a tangent to a circle.
• It can be used while painting and installation of tiles.
• The theorem is used to determine the fare of a journey.
• The basic Proportionality Theorem helps in calculating the dimensions of the land.

---

Things to Remember

• The basic proportionality theorem is used to measure the length of the sides of triangles.
• It states that when a line is drawn parallel to one of the sides of a triangle, the parallel line divides the other two sides in equal proportion.
• The converse of BPT states when a line is drawn to cut, the two sides of a triangle are equal in proportion to the third side.
• The line divides the other two sides of the triangle at its midpoint.
• It was given by the Greek Mathematician Thales, who used the concept of similar triangles.

Also Read: