Is AD the bisector of angle A in triangle ABC?

Question

(1) Triangle ABD and triangle ACD have equal areas
(2) AD is not perpendicular to BC

Guys any idea how to solve this question?

NightFury · Answer

If areas are equal then D is midpoint of B & C.AD will be median.
Bcoz 4r both triangles heights are same.
Median can only be angular bisector when AD is perpendicular to BC.
This can be obtained from using congruence of triangles.

So data is insufficient.

Hope that helps.

vibhav · Answer

Thanks NightFury! makes complete sense! I had figured the 1st part but the median and angle bisector logic was unknown to me. So basically what you are saying is that a median can be the angle bisector only when it is also the altitude (90deg)?

NightFury · Answer

I am sorry the data makes complete sense.
Using both we can determine that AD is not an angular bisector.It stands out just as median.

Yes. A median can be angular bisector if it is an altitude.

vibhav · Answer

thanks nightfury. very helpful!

sandal85 · Answer

Below cases can be possible and can be useful in exam.

1) Median ,Angle bisector and perpendicular will be same if triangle is isosceles or equilateral triangle. Opposite is also true for this.
2) Median can be angle bisector only if it is perpendicular as well and this applies vice-versa like angle bisector can be median if it is perpendicular also.

sandal85 · Answer

Answer of this would be C as we can figure out that AD is not abgle bisector.

harshavmrg · Answer

I could not understand the problem, can somebody help me understand this

vibhav · Answer

Thanks sandal85. Indeed useful corollaries to remember.

vibhav · Answer

Harsha, this is a problem in geometry in which you have a triangle abc. A line AD is made that intersects the line BC at point D. We are asked if this line AD is an angle bisector of angle A.
1. says that the area of are of the two triangles now formed (becuase of line AD) are equal
2. says that this line AD is perpendicular to side BC

does it make sense now?

narangvaibhav · Answer

Thanks Sansal85... these tips really help... keep posting

cyberjadugar · Answer

Why is anyone assuming that point D is on BC?
If it is a point somewhere else, AD may/may not be the median.

Please explain.

vibhav · Answer

cyberjadugar, you are right. It cannot be assumed that D is on line BC unless explicitly stated in some way. However, i this I assume is not the result of poor question framing more than anything! on the GMAT it would be explicitly stated that D is on BC.

vibhav · Answer

However, cyberjadugar, IMO even if we assume that D is not a point on BC the result doesnt change much. Both the statements still continue to provide the same amount of information.

Edvento · Answer

Ideally, the GMAC would surely make sure to mention that D is a point on BC. However, the question is a good one as a number of times we make a mistake of not choosing a possibility that 'NO' can also be an answer in DS.

Regards,

Shouvik.

jlgdr · Answer

But won't AD only be bisector of angle A if D is the height on segment BC hence perpendicular? Could someone please explain this one?

I think what’s going on here.

Statement 1 we have that the areas of both triangles created by line AE have same areas. Well, we still don't know if this is the bisector of angle A. Put it this way, we could have that BC>AC, and thus compensating for the area, AD needs not to be the bisector of A. In other words, it will ONLY be the bisector of A is AD is perpendicular to BC, which we can't assume from this information

Statement 2 tells us that AD is NOT perpendicular to BC. We could have AD be the bisector of angle A (as in statement 1) if the triangle is constructed similarly. Likewise, we could have that AD stands somewhere else thus making the triangle have different areas. Clearly insufficient

Both together, we know that the areas are the same and that AD is NOT perpendicular to BC. Therefore, AD CANNOT be the bisector of angle A for the reasons explained above.

C stands

Thanks
Cheers
J

umg · Answer

Calling Experts..

I don't think OA in this question is correct..

Assuming that D is a point on BC,

(1) Triangle ABD and triangle ACD have equal areas
Means that BD = CD

Because Area =  \frac{1}{2}*Base *Height 
Since, Height is Constant, Bases of both the triangles has to be equal. Hence, CD = BD.
Therefore, AD has to be the Angle Bisector Irrespective of it being Perpendicular to BC.

My Answer: A

martinjudge91 · Answer

Expert solution required. thanks

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Is AD the bisector of angle A in triangle ABC?

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Is AD the bisector of angle A in triangle ABC?

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