Question

what is basic proportionality theorem explain it

Answer 1

Basic Proportionality Theorem and Equal Intercept Theorem

Basic Proportionality Theorem: If a line is parallel to a side of a triangle which intersects the other sides into two distinct points, then the line divides those sides in proportion. Let’s not stop at the statement, we need to find a proof that its true. So shall we begin?





Classification of Triangle



Basic Proportionality Theorem



Pythagoras Theorem

 

Introduction

Basic Proportionality Theorem was first stated by Thales, a Greek mathematician. Hence it is also known as Thales Theorem. Thales first initiated and formulated the Theoretical Study of Geometry to make astronomy a more exact science. What is this theorem that Thales found important for his study of astronomy? Let us find it out.

Basic Proportionality Theorem (can be abbreviated as BPT) states that, if a line is parallel to a side of a triangle which intersects the other sides into two distinct points, then the line divides those sides in proportion. 

 

In the figure alongside, if we consider DE is parallel to BC, then according to the theorem,

ADBD=AECE

Let’s not stop at the statement, we need to find a proof that its true. So shall we begin?

PROOF OF BPT

Given: In  ΔABC, DE is parallel to BC

Line DE intersects sides AB and PQ in points D and E, such that we get triangles A-D-E and A-E-C.

To Prove: ADBD=AECE

Construction: Join segments DC and BE

Proof: 

In ΔADE and ΔBDE,

A(ΔADE)A(ΔBDE)=ADBD                 (triangles with equal heights)

In ΔADE and ΔCDE,

A(ΔADE)A(ΔCDE)=AECE                  (triangles with equal heights)

Since ΔBDE and ΔCDE have a common base DE and have the same height we can say that,

A(ΔBDE)=A(ΔCDE)

Therefore,

A(ΔADE)A(ΔBDE)=A(ΔADE)A(ΔCDE)

Therefore,

ADBD=AECE

Hence Proved.

The BPT also has a converse which states, if a line divides any two sides of a triangle in the same ratio, then the line is parallel to the third side.

(Note: A converse of any theorem is just a reverse of the original theorem, just like we have active and passive voices in English.)

PROPERTIES OF BPT

The BPT has 2 properties.

Property of an angle bisector.Property of Intercepts made by three parallel lines on a transversal.

Property of an Angle Bisector

Statement: In a triangle, the angle bisector divides the side opposite to the angle in the ratio of the remaining sides.



In the given figure, seg AD is the angle bisector of ∠BAC.

According to the property,

BDDC=ABAC

Property of Intercepts made by three parallel lines on a transversal

Statement: The ratio of the intercepts made on the transversal by three parallel lines is equal to the ratio of the corresponding intercepts made on any other transversal of the same parallel line.



Consider the above figure,  line l, m and n are parallel to each other. Transversals p and q intersect the lines at point A, B, C and D, E, F. So according to the property,

ABBC=DEEF

solve example

Q: In triangle ABC, seg AD is the angle bisector of∠BAC. BD=6, DC=8, AB=15 Find AC



Solution:

segments AD bisects ∠BAC     (given)

.’. AB/AC=BD/DC                      (Angle bisector property)

Assume AC= x

.’. 15/x=6/8

.’. 15 × 8 =6 × x

.’.  x= (15 × 8) / 6 = 20.

Therefore, AC= 20.

Q: We are given that in triangle PQR, MN intersects PQ and PR at M and N respectively such that PM = 3 cm, MQ = 9 cm, PN = 2 cm and NR = 6 cm. Is MN parallel to QR?

Solution:

PM= 3 cm, MQ=9  cm

Now, PM/MQ=3/9=1/3

Also, PN=2 cm, NR=6 cm.

Now, PN/NR= 2/6=1/3

Therefore, PM/MQ=PN/NR

Hence, by the converse of basic proportionality theorem, we have MN parallel to QR.

Answer 2

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