Question

in triangle ABC altitude BE and CF are equal prove that the triangle is isosceles

Answer 1

Given : AD = BE = CF and angles A = B = C= 90°

To prove : Triangle ABC is equialteral.

Proof : In triangle ABE and ACF

Angle AEB = AFC ( 90° each )

BE = CF ( Given )

Angle A = A ( Common )

Hence Triangle ABE ~ ACF by AAS congruency

AB = AC ( c.p.c.t )

In triangle AOE and EOC

OE = OE ( Common )

Angle E = E ( 90° each )

AO = OC [ :. AD = FC, their halfs OA = OC ]

Hence triangle AOE ~ EOC by RHS congruency

AE = EC ( c.p.c.t )

In triangle ABE and BCE

Angle E = E ( 90° each )

BE = BE ( common )

AE = EC ( proved above )

Hence, triangle ABE ~ BCE by SAS congruency.

AB = BC ( c.p.c.t )

As AB = AC and AB = BC, so AC = BC.

Hence, AB = BC = CA.

HENCE PROVED.Answer:

Step-by-step explanation:

Answer 2

Hello mate ^_^

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Solution:

In ∆BEC and ∆CFB

BE=CF                (Given)

∠BEC=∠CFB              (Each given equal to 90°)

BC=CB                (Common)

Therefore, by RHS rule, ∆BEC≅∆CFB

It means that ∠C=∠B        (Corresponding parts of congruent triangles are equal)

⇒AB=AC                (In a triangle, sides opposite to equal angles are equal)

Therefore, ∆ABC is isosceles.

hope, this will help you.

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