BPT THEOREM forward and backward statement and derive 5 more results from given result (AD/DB)=(AE/EC)
Answers
Answer
In △ABC, we have
DE||BC
⇒ ∠ADE=∠ABC and ∠AED=∠ACB [Corresponding angles]
Thus, in triangles ADE and ABC, we have
∠A=∠A [Common]
∠ADE=∠ABC
and, ∠AED=∠ACB
∴ △AED∼△ABC [By AAA similarity]
⇒
AB
AD
=
BC
DE
We have,
DB
AD
=
4
5
⇒
AD
DB
=
5
4
⇒
AD
DB
+1=
5
4
+1
⇒
AD
DB+AD
=
5
9
⇒
AD
AB
=
5
9
⇒
AB
AD
=
9
5
∴
BC
DE
=
9
5
In △DFE and △CFB, we have
∠1=∠3 [Alternate interior angles]
∠2=∠4 [Vertically opposite angles]
Therefore, by AA-similarity criterion, we have
△DFE∼△CFB
⇒
Area(△CFB)
Area(△DFE)
=
BC
2
DE
2
⇒
Area(△CFB)
Area(△DFE)
=(
9
5
)
2
=
81
25
.[Using (i)]
solution
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Given : A Δ ABC and a line intersecting AB in D and AC in E, such that AD / DB = AE / EC. Prove that : DE || BC Let DE is not parallel to BC.