Triangle Proportionality Theorem

The geometric figures which have the same shape but different dimensions are known as similar figures.

While two geometric figures which have the same shape and size are known as congruent figures.

Two congruent figures are always similar but two same figures need not be congruent.

For example, all the circles of different or same radii are similar to each other. Similarly all the rectangles, squares, etc are similar to each other.

For two triangles to be similar two conditions need to be fulfilled

• their corresponding angles are equal, and

• their corresponding sides are proportional    

We can put it this way, the two triangles ∆ABC and ∆PQR will be similar when,

i) ∠A = ∠P, ∠B = ∠Q and ∠C = ∠R

ii) \[\frac{AB}{PQ}\] = \[\frac{BC}{QR}\] = \[\frac{AC}{PR}\]

Two polygons are said to be similar if 

• their corresponding angles are equal

• their corresponding sides are proportional. 

We'll now discuss an important theorem which is a result of similar triangles known as triangle proportionality theorem or proportionality theorem.

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

                                          (image will be uploaded soon)

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 

                                          (image will be uploaded soon)

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.