Proportionality Theorem or Thales Theorem
The basic proportionality theorem was given by Thales. It states: “A line drawn parallel to one side of a triangle to intersect the other two sides in distinct points divides the other two sides in the same ratio”. Consider a triangle ABC given below, in this triangle we draw PQ||BC, now according to triangle proportionality theorem the ratio of AP to PB will be equal to the ratio of AQ to QC I.e. AP/PB=AQ/QC
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Let us now prove the theorem.
Given: A triangle ABC in which PQ||BC and PQ intersects AB and AC at P and Q respectively.
To prove: AP =AQ
PB. QC
Construction: Join BQ and CP.
Draw QN\[\perp\]AB and PM\[\perp\]AC
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Proof : we have :
ar(∆APQ)= 1/2 ×AP×QN. [Area of triangle=(base×height)÷2]
ar(∆PBQ)= 1/2 ×PB×QN
ar(∆APQ)=1/2×AP×QN …..(1)
ar(∆PBQ). 1/2×PB×QN
Again,ar(∆APQ)=1/2 ×AQ×PM=ar(∆AQP)
ar(∆QCP)=1/2 ×QC×PM
Therefore, ar(∆APQ)=1/2×AQ×PM……(2)
ar(∆QCP). 1/2×QC×PM
Now,∆PBQ and ∆QCP being on the same base PQ and between the same parallels PQ and BC, we have:
ar(∆PBQ)=ar(∆QCP)......(3)
From (1),(2) and (3), we have:
AP/PB=AQ/QC
Hence the triangle proportionality theorem is proved. Also, ∆ABC and ∆APQ satisfy the required conditions for similar triangles as stated above. Therefore, it can be concluded that ∆ABC ~∆APQ.
Example 1: In a triangle ABC,DE||BC.If AD=2.5cm,DB=3cm and AE=3.75cm,find AC.
In ∆ABC,DE||BC
Therefore, AD/DB=AE/EC (by thales' theorem)
⇒ 2.5/3=3.75/x,where EC=x cm
⇒ x=(3×3.75)/2.5=9/2=4.5
⇒ EC=4.5 cm
Hence, AC=(AE+EC)=(3.75+4.5)=8.25 cm.
Converse of Thales Theorem
If two sides of a triangle are divided in the same ratio by a line then the line must be parallel to the third side.
Given: A triangle ABC and a line l intersecting AB at D and AC at E, such that AD/DB=AE/EC.
To prove: DE||BC
Proof: If possible let not be parallel to BC. Then there must be another line through D, which is parallel to BC.Let DF||BC
AD/DB=AF/FC……(1)
But,AD/DB=AE/EC(given)......(2)
From (1) and (2),we get:
AF/FC=AE/EC
⇒AF/FC+1=AE/EC+1
⇒(AF+FC)/FC=(AE+EC)/EC
⇒AC/FC=AC/EC
⇒1/FC=1/EC
⇒FC=EC.
This is possible only when E and F coincide.
Hence DE||BC.
Example2:
If D and E are points on the sides AB and AC respectively of ∆ABC such that AB=5.6cm ,AD=1.4cm,AC=7.2cm and AE=1.8cm,show that DE||BC.
Given: AB=5.6cm ,AD=1.4cm,AC=7.2cm and AE=1.8cm
Therefore AD/AB=1.5/5.6=1/4and AE/AC=1.8/7.2=¼
⇒AD/AB=AE/AC
Hence,by the converse of Thales' theorem DE||BC.
Corollary
In a ∆ABC, a line DE is drawn such that DE||BC and it intersects AB in D and AC in E, then
1)AB/DB=AC/EC
2)AD/AB=AE/AC
Fun Facts About Thales Theorem
1. Thales fell into a well and noticed that he became interested in the stars. He could not even see what was before his feet because he was so curious to know about what was going on in heaven.
2. None of his writing services have been found. This is why it is hard to determine his philosophy.
3. He is famous for his theory of oil presses.
4. Five theorems of elementary geometry have been given by him.
5.As quoted by Thales... "Space is the greatest thing, it contains all things."
6. He predicted a solar eclipse accurately.
7. Predicted a good harvest season and then bought all olive mills.
8. The Sun produces so much energy, that every second the core releases the equivalent of 100 billion nuclear bombs.
9. There are 7 types of different stars.
10. A star is said to born once nuclear fusion starts in its core.
FAQs on Triangle Proportionality Theorem
1. What is the Triangle Proportionality Theorem, also known as the Basic Proportionality Theorem (BPT)?
The Triangle Proportionality Theorem, also commonly known as the Basic Proportionality Theorem (BPT) or Thales's Theorem, states that if a line is drawn parallel to one side of a triangle and intersects the other two sides at distinct points, then it divides the two sides in the same ratio. For a triangle ABC, if a line DE is parallel to BC and intersects sides AB and AC at D and E respectively, then AD/DB = AE/EC.
2. What is the formula for the Triangle Proportionality Theorem?
The theorem is expressed as a ratio. In a triangle ΔABC, if a line segment DE is drawn parallel to the side BC such that it intersects sides AB and AC at points D and E, respectively, the formula representing the proportional sides is:
AD/DB = AE/EC.
This formula is fundamental for solving problems where parts of a triangle's sides are unknown.
3. How is the Triangle Proportionality Theorem used to find an unknown side length?
To find an unknown length, you can set up an equation using the theorem's proportion. For instance, if you are given a triangle ABC with a line DE parallel to BC, and you know the lengths of AD, DB, and AE, you can find EC. By arranging the formula AD/DB = AE/EC, you can substitute the known values and solve for the unknown side length, often represented by a variable like 'x'.
4. What is the statement and importance of the converse of the Triangle Proportionality Theorem?
The converse of the Triangle Proportionality Theorem states that if a line divides any two sides of a triangle in the same ratio, then the line must be parallel to the third side. Its main importance is as a method to prove that two lines are parallel. If you can demonstrate that AD/DB = AE/EC in a triangle ABC, you can conclude that the line segment DE is parallel to the side BC.
5. Can you provide a simple example of the Triangle Proportionality Theorem in action?
Certainly. Imagine a triangle PQR where a line ST is parallel to side QR. The line ST intersects side PQ at S and side PR at T. If the lengths are given as PS = 3 cm, SQ = 6 cm, and PT = 4 cm, we can find the length of TR.
Using the theorem:
PS/SQ = PT/TR
3/6 = 4/TR
1/2 = 4/TR
Therefore, TR = 8 cm. This shows how the theorem helps find missing lengths based on proportionality.
6. How is the proof of the Basic Proportionality Theorem derived?
The proof of the Basic Proportionality Theorem is typically derived using the formula for the area of a triangle (Area = ½ × base × height). The key steps are:
- Consider a triangle ABC with DE parallel to BC.
- Calculate the ratio of the areas of ΔADE and ΔBDE, which simplifies to AD/DB.
- Similarly, calculate the ratio of the areas of ΔADE and ΔCDE, which simplifies to AE/EC.
- Recognise that ΔBDE and ΔCDE have the same area because they are on the same base (DE) and between the same parallel lines (DE and BC).
- By equating the area ratios, we prove that AD/DB = AE/EC.
7. What is the core difference between the Triangle Proportionality Theorem and the Midpoint Theorem?
The key difference lies in their specificity. The Midpoint Theorem is a special case of the Triangle Proportionality Theorem.
- The Triangle Proportionality Theorem applies when a line parallel to a side divides the other two sides in any proportion.
- The Midpoint Theorem applies only when the line connects the midpoints of the two sides, meaning the proportion is exactly 1:1. It also adds a conclusion that the line segment is half the length of the third side, a property not covered by the general Proportionality Theorem.
8. How does the Triangle Proportionality Theorem relate to the concept of similar triangles?
The Triangle Proportionality Theorem is a direct consequence of triangle similarity. When a line DE is drawn parallel to BC in a triangle ABC, it creates a smaller triangle, ΔADE. This smaller triangle is similar to the larger triangle ΔABC (by Angle-Angle similarity criterion, as ∠ADE = ∠ABC and ∠AED = ∠ACB). Because the triangles are similar, the ratio of their corresponding sides is equal, which leads directly to the proportional relationship described by the BPT.
9. Are there any practical, real-world applications of the Triangle Proportionality Theorem?
Yes, the theorem is a foundational principle in various fields. In architecture and engineering, it is used to create scaled drawings and models, ensuring that all parts of a structure maintain their correct proportions. In surveying and cartography (map-making), it helps in calculating distances and heights that cannot be measured directly by using proportional relationships in triangles formed by lines of sight.
