Question

In ∆ ABC, the midpoints of AB and AC are D and E respectively. CF is parallel to AB and meets DE produced at F. Prove that :
(a) DB = FC
(b) DE = EF​

Answer 1

Step-by-step explanation:

Given :-

In ∆ ABC, the midpoints of AB and AC are D and E respectively.

CF is parallel to AB and meets DE produced at F.

Required To Prove:-

(a) DB = FC

(b) DE = EF

Proof :-

In ∆ ABC, the midpoints of AB and AC are D and E respectively.

CF is parallel to AB and meets DE produced at F.

In ∆ ADE and ∆ CFE

AE ≅ EC ( E is the mid point of AC)

∠AED ≅ ∠ CEF (Vertically opposite angles)

∠ADE ≅ ∠ CFE ( Alternative interior angles as CF || BA with transversal DF)

BY ASA Congruency rule,

Therefore, ∆ ADE ≅ ∆ CFE

Thus AD ≅ CF -----------(1)

and DE ≅ EF --------------(2)

Since, Corresponding parts are congruent in the congruent triangles .

We know that

E is the mid point of AC

D is the mid point of AB.

Therefore, AD = DB -----------(3)

From (1) & (3)

AD = DB = FC

Therefore, DB ≅ FC

Therefore,

DB = FC and DE = EF

Hence, Proved.

Answer 2

Step-by-step explanation:

Step-by-step explanation:

Given :-

In ∆ ABC, the midpoints of AB and AC are D and E respectively.

CF is parallel to AB and meets DE produced at F.

Required To Prove:-

(a) DB = FC

(b) DE = EF

Proof :-

In ∆ ABC, the midpoints of AB and AC are D and E respectively.

CF is parallel to AB and meets DE produced at F.

In ∆ ADE and ∆ CFE

AE ≅ EC ( E is the mid point of AC)

∠AED ≅ ∠ CEF (Vertically opposite angles)

∠ADE ≅ ∠ CFE ( Alternative interior angles as CF || BA with transversal DF)

BY ASA Congruency rule,

Therefore, ∆ ADE ≅ ∆ CFE

Thus AD ≅ CF -----------(1)

and DE ≅ EF --------------(2)

Since, Corresponding parts are congruent in the congruent triangles .

We know that

E is the mid point of AC

D is the mid point of AB.

Therefore, AD = DB -----------(3)

From (1) & (3)

AD = DB = FC

Therefore, DB ≅ FC

Therefore,

DB = FC and DE = EF

Hence, Proved.

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