state n prove Thales theorem
Answers
in geometry ,Thales theorem states that if A and B ,C are distinct point on a circle where the line Ac is a diameter ,then the angle ABC is a right angle....
THALES THEOREM OR BASIC PROPORTIONALITY THEOREM
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,
A D
B D
=
A E
C E
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 AC in points D and E respectively.
To Prove:
A D
B D
=
A E
C E
Construction:
Draw EF ⟂ AD and DG⟂ AE and join the segments BE and CD.
Proof:
Area of Triangle= ½ × base× height
In ΔADE and ΔBDE,
A r ( A D E )
A r ( D B E )
=
1
2
×AD×EF
1
2
×DB×EF=
AD
DB
(1)
In ΔADE and ΔCDE,
A r ( A D E )
A r ( E C D )
=
1
2
×AE×DG
1
2
×EC×DG=
A E
E C
(2)
Note that ΔDBE and ΔECD have a common base DE and lie between the same parallels DE and BC. Also, we know that triangles having the same base and lying between the same parallels are equal in area.
So, we can say that
Ar(ΔDBE)=Ar(ΔECD)
Therefore,
A(ΔADE)
A(ΔBDE)
=
A(ΔADE)
A(ΔCDE)
Therefore,
A D
B D
=
A E
C E
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.)