Question

AD ,BE, CF are the altitude of triangle ABC and are equal .prove that triangle ABC is equilateral Triangle .​

Answer 1

Answer:

Step-by-step explanation:

Given :-

A ΔABE and ΔACF

To Prove :-

ABC is an equilateral triangle.

Proof :-

In ΔABE and ΔACF

BE = CF (given)

∠A = ∠A (common)

∠AFB = ∠AFC = 90°

AB = AC  

(Therefore Triangle DAC is congruent to Triangle CBE)

AC = BC  

Triangle BCF is congruent to Triangle ABD

 AB = BC

AC = BC

AB = AC

AB = BC = AC  

Hence, ABC is an equilateral triangle using congruence.

Answer 2

Answer:

The ABC triangle is a equaliteral triangle

Proved.

Step-by-step explanation:

Given Data -

• AD ,BE, CF are the altitude of triangle ABC.
• AD ,BE, CF are equal.

To Prove -

The Triangle ABC is equilateral Triangle.

Proof -

So,

In triangle ABE and triangle ACF.

As given, BE = CF

⇒ ∠A = ∠A [ Common ]

So,

⇒ ∠AEB = ∠AFC = 90°

Triangle DAC is congruent to Triangle CBE.

We know that -

The AAS Rule.

Statement -

Two triangle are congruent if any two pair of angles and one pair of corresponding sides are equal.

So,

It implies,

⇒ ABE ≅ ACF.

Now,

⇒ AB = AC

Same thing here,

BCF ≅ ABD

⇒ AB = BC

⇒ AC = BC

⇒ AB = AC

⇒ AB = BC = AC

_______________[ Proved ]