Question

if altitude of a triangle are equal then prove that the given triangle is an equilateral triangle​

Answer 1

Answer:

Step-by-step explanation:

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.