Triangle DS

[knight247]

Don't have an OA on this one.

[neelgandham]

Is Triangle ABC Isosceles ?

1) The area of triangular region ABD is equal to area of triangular region DBC

Area of triangular region ABD = Area of triangular region DBC
(1/2)* height *AD = 1/2*height *DC
AD = DC (Since height of both the triangles is equal)
Still insufficient!

2)BC_|_AC and AD = DC

The Triangles ABD and CBD are congruent(Side-Angle-Side congruence)
BC = BC
AD = DC
Angle BDA = Angle BDC = 90
So AB = BC. But we still don't know the value of the length of side AC.
If AC!=AB(=BC), then Triangle ABC is Isosceles
If AC=AB=BC, then Triangle ABC is Equilateral.
Insufficient!

IMO E

[shankar.ashwin]

Is Triangle ABC Isosceles ?

1) The area of triangular region ABD is equal to area of triangular region DBC

Area of triangular region ABD = Area of triangular region DBC
(1/2)* height *AD = 1/2*height *DC
AD = DC (Since height of both the triangles is equal)
Still insufficient!

2)BC_|_AC and AD = DC

The Triangles ABD and CBD are congruent(Side-Angle-Side congruence)
BC = BC
AD = DC
Angle BDA = Angle BDC = 90
So AB = BC. But we still don't know the value of the length of side AC.
If AC!=AB(=BC), then Triangle ABC is Isosceles
If AC=AB=BC, then Triangle ABC is Equilateral.
Insufficient!

IMO E

But aren't all equilateral triangles also isosceles ?

[neelgandham]

But aren't all equilateral triangles also isosceles ?

Aw! Thanks for correcting me. I always thought Isosceles triangle is a triangle with two equal sides and two only. but mathwiki says "An isosceles triangle is a triangle with (at least) two equal sides.An equilateral triangle is therefore a special case of an isosceles triangle having not just two, but all three sides and angles equal.".

Source: https://math.wikia.com/wiki/Isosceles_triangle

Answer should be B