Question

ABC is a triangle with D as mid point of AB. DE is perpendicular to AC and DF and BC. Also DE = DF. Prove that Δ ABC is an isosceles Δ .

Don't give wrong answer.
Otherwise I report that answer.​

Answer 1

Given :

• ABC is a triangle with D as mid point of AB.
• DE is perpendicular to AC and DF and BC.
• Also DE = DF.

To prove :

• Prove that Δ ABC is an isosceles Δ .

Solution :

In triangle DEB and triangle DFC ,

• Angle E = Angle F ( each 90° )
• DE = DF ( given )
• BD = DC ( D is the midpoint )

Therefore triangle DEB and triangle DFC are congruent by SAS congruency criteria .

Now ,

Angle B = Angle C by CPCT - 1

From equation 1 we get ,

AB = AC ( As sides opposite to equal angles are equal )

So as the two sides of traingle ABC are equal . Thus , proved that it is an isosceles triangle

More to know :

A triangle is an enclosed figure with three sides , vertices and angles .

CPCT is corresponding parts of congruent triangles . That states that if two traingles are congruent then their corresponding parts will be equal .

Theorem used :

Sides opposite to equal angles are equal

SAS criteria :

Side angle side

If two sides and one angle of two traingles are equal then they are congruent

If a point is given as midpoint then it divides the line segment into two equal parts .

An isosceles triangle has two sides equal

Answer 2

$\gray{ \tt \: answer}$

Given :

ABC is a triangle with D as mid point of AB.

DE is perpendicular to AC and DF and BC.

Also DE = DF.

To prove :

Prove that Δ ABC is an isosceles Δ .

Solution :

In triangle DEB and triangle DFC ,

Angle E = Angle F ( each 90° )

DE = DF ( given )

BD = DC ( D is the midpoint )

Therefore triangle DEB and triangle DFC are congruent by SAS congruency criteria .

Now ,

Angle B = Angle C by CPCT - 1

From equation 1 we get ,

AB = AC ( As sides opposite to equal angles are equal )

So as the two sides of traingle ABC are equal . Thus , proved that it is an isosceles triangle

More to know :

A triangle is an enclosed figure with three sides , vertices and angles .

CPCT is corresponding parts of congruent triangles . That states that if two traingles are congruent then their corresponding parts will be equal .

Theorem used :

Sides opposite to equal angles are equal

SAS criteria :

Side angle side

If two sides and one angle of two traingles are equal then they are congruent

If a point is given as midpoint then it divides the line segment into two equal parts .

An isosceles triangle has two sides equal