what is BPT theorem?
Answers
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.
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.
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.)
Read the properties of Triangles and Quadrilaterals here.
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.
Angle bisector
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.
intercepts
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
Answer:
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here is your answer
BASIC PROPORTIONALITY THEOREM (bpt)
states that if a line is drawn parallel to one side of a triangle to interest the other two sides in two distinct points then the other two sides are divided in the same ratio...
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