Q. 25
State and prove thales theorem.
Answers
Answer:
Step-by-step explanation:
Basic Proportionality Theorem ( 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.
Basic Proportionality Theorem
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 AC in points D and E respectively.
To Prove: ADBD=AECE
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,
Ar(ADE)Ar(DBE)=12×AD×EF12×DB×EF=ADDB(1)
In ΔADE and ΔCDE,
Ar(ADE)Ar(ECD)=12×AE×DG12×EC×DG=AEEC(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,
ADBD=AECE
Hence Proved.
Answer:
Step-by-step explanation:
Thales theorem or BPT