state and prove BPT theorem using that theorem
find the length of AE,
AD = 1.8cm, BD=5.4cmand
EC=7.2
Answers
Answer:
7.2×1.8÷2.5
Step-by-step explanation:
AD/DB=AE/EC
hi frnd
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.

In the given figure, seg AD is the angle bisector of ∠BAC.
According to the property,
BD