Question

please solve it......​

Answer 1

Given :

• ABC is an isosceles triangle.

• AB = AC

• AE = EC , AF = BF

To prove :

• BE = CF

Proof :

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ☯ In Δ ABE and Δ ACF ,

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀

$: \implies \bf \: AF = AE \\ \\ \\ \\ : \implies \: \bf \: AB = AC \\ \\ \\ \\ :\implies \bf{\dfrac{{AB}}{{2}}} \: = {\dfrac{{AC}}{{2}}} \\ \\ \\ \\ :\implies \bf \: AF = AE \\ \\ \\ \\ :\implies \bf \: AB = AC \: \: (Given) \\ \\ \\ \\ :\implies \bf \: ∠ BAE = ∠ CAF \: (Common) \: \\ \\ \\ \\ \: :\implies \bf \: \: ∠ ABE \: ≅ Δ ACF \: (S.A.S) \\ \\ \\ \\ :\implies \bf \: \: BE = CF \: (C.P.C.T)$

Hence, solved.

Answer 2

$\huge\bf{Concept :}$

• Here, we're asked to prove that BE = CF. For proving the given equation, we must prove the congruence of a pair of triangles, and then prove the equation by Corresponding Part Of Congruent Triangles.
• So, we'll take ∆ACF and ∆ABE and then prove BE = CF.

Why we're choosing ∆ACF and ∆ABE?

We're choosing these two Triangles because the sides which is included in the equation to be proved is there, i.e. sides BE and CF.

Now, we're in the position to answer the given question!

$\huge\bf{Given :}$

• ABC is an isosceles triangle with AB = AC.
• F and E are the mid-points of the sides AB and AC respectively, i.e AE = CE and AF = BF.

$\huge\bf{To\: Prove :}$

• BE = CF

$\huge\bf{Proof :}$

In ∆ACF and ∆ABE,

⠀⠀⠀⠀⠀⠀⠀⠀⠀AB = AC [ Given ] --------(i)

• Dividing both sides by 2.

⠀⠀⠀⠀⠀⠀⠀⠀⠀AB/2 = AC/2

⇒ ⠀⠀⠀⠀⠀⠀⠀AF = AE ----------(ii)

• [ Since, F and E are the mid-points of AB and AC. Therefore, AB/2 = AF and AC/2 = AE ]

⠀⠀⠀⠀⠀⠀⠀⠀⠀∠CAF = ∠BAE [ Common Angle ]-------(iii)

From (i), (ii) and (iii)

⠀⠀⠀⠀⠀⠀⠀⠀⠀∆ACF ≅ ∆ABE [ By SAS congruency ]

∴ BE = CF [ By C•P•C•T ]

Hence, proved.