MGMAT Geometry Question Bank

[nadontheway]

Why is A not sufficient??

From A we know that ACD is isoceles right triangle with hypothenuse AD.
Angle CAD is 45° and angle ACD is 90° so CDA must be 45°.
CD should be 5/Sqroot2.
And BD should be 2*5/Sqroot2

Why am i wrong?

Thx

[Aragorn]

Ist condition doesn't say isoscles

[nadontheway]

although the statement A doesn't say isoceles, you can deduce it thanks to the statement itself.
The perpendicular cuts the right triangle into two triangles creating a right angle (ACD) and splitting the right triangle ABD in two (45°). So you can deduce that angle ADC is 45°.
So A should be sufficient. Am I clear?

[nadontheway]

Hi

can Mgmat staff help me with this question on hold?

much appreciated.

[rfernandez]

although the statement A doesn't say isoceles, you can deduce it thanks to the statement itself.
The perpendicular cuts the right triangle into two triangles creating a right angle (ACD) and splitting the right triangle ABD in two (45°). So you can deduce that angle ADC is 45°.

It's not necessarily true that the altitude AC bisects angle BAD. Consider, for example, if triangle ABD were a 30-60-90 right triangle such that angle D's measure is 60 and angle B's measure is 30. Drawing an altitude from angle A to opposite side BD at point C would split angle BAD into angle BAC with measure 60 and angle CAD with measure 30. So it's assuming too much to say that the altitude AC necessarily bisects angle BAD.

It would be true that altitude AC bisects angle BAD if and only if triangle ABD were a 45-45-90 right triangle to begin with, but this was not given.

Incidentally, what we can infer from (1) is that three similar triangles are created: triangle ABD ~ triangle CAD ~ triangle CBA. That tells us that corresponding side lengths are proportional, but with only one side measure (AD=5) we can't make any conclusions about the length of BD.