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Basic Proportionality Theorem (BPT) | Proof and Examples

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Thales theorem is one of the most fundamental theorems in geometry that relates the parts of the length of sides of triangles. The other name of the Thales theorem is Basic Proportionality Theorem or BPT.

Let’s learn about the Thales Theorem or Basic Proportionality Theorem in detail, including its statement, proof, and converse as well.

Thales Theorem Statement

Thales’s Theorem or Basic Proportionality Theorem (BPT) states that if a line is drawn parallel to one side of a triangle intersecting the other two sides in distinct points, then the other two sides are divided in the same ratio.

Basic Proportionality Theorem Proof

Let’s prove the Basic Proportionality Theorem.

Given: Let us suppose we have a triangle ABC, if we draw a line LM parallel to side BC.

To Proof: \frac{AL}{LB}=\frac{AM}{MC}                  

Basic Proportionality Theorem

Construction

To prove the required result, construct the following lines in the given figure:

  • Construct LY⊥AM.
  • Construct MX⊥AL.
  • Join points L with C and M with B with a line segment.
Construction for Basic Proportionality Theorem

Proof

Since, area of triangle= (\frac{1}{2}\times base\times height)

Area of \triangle ALM=(\frac{1}{2}\times AL\times MX)

Area of \triangle LBM=(\frac{1}{2}\times LB\times MX)

Area of \triangle ALM=(\frac{1}{2}\times AM\times LY)

Area of \triangle LMC=(\frac{1}{2}\times MC\times LY)

Ratio of area of \triangle ALM and \triangle LBM:

\frac{area(\triangle ALM)}{area(\triangle LBM)}=\frac{(\frac{1}{2}\times AL\times MX)}{(\frac{1}{2}\times LB\times MX)}=\frac{AL}{LB}. . .(1)

Ratio of area of \triangle ALM and  LMC:

\frac{area(\triangle ALM)}{area(\triangle LMC)}=\frac{(\frac{1}{2}\times AM\times LY)}{(\frac{1}{2}\times MC\times LY)}=\frac{AM}{MC}. . .(2)

According to the property of triangles, the triangles on the same base and between same parallel lines have equal areas.

Therefore, \triangle LBM and \triangle LMC have equal areas. 

i.e., area of \triangle  LBM = area of \triangle LMC. . .(3)

From equations (1),(2), and (3) we can conclude:

\bold{\frac{AL}{LB}=\frac{AM}{MC}}  [Hence Proved]

Corollary of Thales Theorem

Mid Point Theorem

Mid Point theorem states that, “If the line is drawn from one of the midpoints of the side of a triangle parallel to the second, then it always intersects the third side at the midpoint as well.”

Learn More:

Converse of Basic Proportionality Theorem

The Converse of the Basic Proportionality Theorem (BPT) or Thales’s Theorem states that if a line intersects two sides of a triangle and divides the sides proportionally, then the line is parallel to the third side of the triangle.

Proof

Let’s prove the converse of the Basic Proportionality Theorem or converse of Thales Theorem is given as follows:

Given: Let us consider a triangle ABC with D and E two points on the side AB and AC such that AD/BD = AE/CE . . .(i).

 Converse of Basic Proportionality Theorem

To Prove: DE∥BC

Construction

To prove the converse of BPT, draw a line DE’∥BC.

Construction for Converse of Basic Proportionality Theorem

Proof

Now, as DE’∥BC in triangle ABC, 

Thus, by using the Basic proportionality theorem,

AD/BD = AE’/CE’ . . .(ii)

Using equations (i) and (ii), we get

AE/CE = AE’/CE’

Add 1 to both sides of the equation,

AE/CE + 1 = AE’/CE’ + 1

⇒ (AE+CE)/CE = (AE’+CE’)/CE’

⇒ AC/CE = AC/CE’

⇒ CE = CE’

This can only be possible if E and E’ are coincident.

Thus, E and E’ are the same points. 

As, DE∥BC 

So, DE∥BC [Hence Proved]

Solved Examples on BPT

Let’s solve some questions on the Basic Proportionality Theorem.

Example 1: In ΔPQR, X and Y are points on the sides PQ and PR respectively such that XY|| QR. If PX/XQ = 2/5 and PR = 10 cm find PY. 

Solution:

In the given triangle PQR, X and Y are points on the sides PQ and PR respectively such that XY || QR.

By the Basic Proportionality Theorem, we have:

PX/XQ = PY/YR

⇒ PX/XQ = PY/(PR-PY)

Given that PX/XQ = 2/5, we can substitute for PX/XQ:

2/5 = PY/(PR-PY)

⇒ 2/5 = PY/(10-PY)

⇒ 2/5 = PY/(10-PY)

⇒ 2(10 – PY) = 5PY

⇒ 20 – 2PY = 5PY

⇒ 7PY = 20 

⇒ PY = 20 /7

Hence, the length of PY is 20/7 cm.

Example 2: In ΔPQR, X and Y are points on the sides PQ and PR respectively such that XY|| QR. If PX = x − 7 , XQ = x + 3 , PY = 2x and YR = 2x −1 , find the value of x.

Solution:

In the given triangle PQR, X and Y are points on the sides PQ and PR respectively such that XY || QR.

By the Basic Proportionality Theorem, we have:

PX/XQ = PY/YR

⇒ (x – 7)/(x + 3) = 2x/(2x – 1)

⇒ (x – 7)(2x – 1) = 2x(x + 3)

⇒ 2x2 – x – 14x +7 = 2x2 + 6

⇒ -15x = 6 – 7

⇒ 15x = 1

⇒ x = 1/15

Example 3: In an ∆ABC, sides AB and AC  are intersected by a line at D and E respectively, which is parallel to side BC. Prove that AD/AB = AE/AC.

Solution:

Given: DE || BC. So, AD/DB = AE/EC

or By interchanging the ratios as => DB/AD = EC/AE

Now, add 1 on both sides 

(DB/AD) + 1 = (EC/AE) + 1

⇒ (DB + AD)/AD = (EC + AE)/AE

⇒ AB/AD = AC/ AE

If we interchange the ratios again, we get

AD/AB = AE/AC [Hence Proved]

Example 4. In triangle ABC, where DE is a line drawn from the midpoint of AB and ends midpoint of AC at E. AD/DB = AE/EC and ∠ADE = ∠ACB. Then prove ABC is an isosceles triangle.

Solution:

Given: AD/DB = AE/EC

By the converse of the basic proportionality theorem, we get => DE || BC

According to question; ∠ADE = ∠ACB

Hence, ∠ABC = ∠ACB

The side opposite to equal angles is also equal to AB = AC

Hence, ABC is an isosceles triangle.

FAQs on Basic Proportionality Theorem

What is Basic Proportionality Theorem?

In simple words, Basic Proportionality Theorem states that the line joining the points on the sides of triangle if parallel then always divide the sides in equal proportions.

Why is Basic Proportionality Theorem called Thales Theorem?

The basic proportionality theorem (BPT) is called Thales Theorem as it is given by the Greek Mathematician cum philosopher Thales of Miletus.

Can Basic Proportionality Theorem (BPT) be applied to any triangle?

Yes, Basic Proportionality Theorem can be applied to any triangle regardless of its size and shape.

Can Thales Theorem be applied if the line joining the sides is not parallel to the base of triangle?

No, the Thales Theorem can only be applied if the line joining the sides of a triangle is parallel as it is the necessary condition for the theorem.

What is converse of Basic Proportionality Theorem?

Converse of the Basic Proportionality Theorem states that if a line intersects two sides of a triangle and divides the sides proportionally, then the line is parallel to the third side of the triangle.



Last Updated : 29 Feb, 2024
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