Triangles

[Acecustis]

If AD is 6√3, and ADC is a right angle, what is the area of triangular region ABC?

(1) Angle ABD = 60°

(2) AC = 12

https://cdn2.manhattanprep.com/images/g ... gleabc.jpg

A. Statement 1 is sufficient
B. Statement 2 is sufficient
C. Both statements together are sufficient
D. Both statements alone are sufficient
E. Neither statement is sufficient alone or together

I answered D and the correct answer is C.

My reasoning:

(1) If angle ABD = 60 degrees then this is a 30-60-90 triangle. Therefore the ratio is 6 : 6√3 : 12. Since AD bisects the triangle I naturally assumed that the total base (BC) would be 12.

(2) If AC = 12 and AD = 6√3, I can use Pyth. Theorem and get b = 6. Since AD bisects the triangle I naturally assumed that the total base (BC) would be 12.

Now in the explanations it just mentions that I only know one side of the split triangle and calls each statement alone insufficient. So my question is what mistake am I making? The question stem doesn't say AD bisects the triangle but wouldn't I naturally assume this because it gives me a right angle? What am I missing here?

Any help is appreciated.

 

[Dr.Peterson]

My reasoning:

(1) If angle ABD = 60 degrees then this is a 30-60-90 triangle. Therefore the ratio is 6 : 6√3 : 12. Since AD bisects the triangle I naturally assumed that the total base (BC) would be 12.

(2) If AC = 12 and AD = 6√3, I can use Pyth. Theorem and get b = 6. Since AD bisects the triangle I naturally assumed that the total base (BC) would be 12.

Now in the explanations it just mentions that I only know one side of the split triangle and calls each statement alone insufficient. So my question is what mistake am I making? The question stem doesn't say AD bisects the triangle but wouldn't I naturally assume this because it gives me a right angle? What am I missing here?

How do you know that AD bisects the triangle? Are you assuming from the picture that the triangle appears to be isosceles?

In general an altitude of a triangle (AD) is not also a perpendicular bisector; this is only true if the triangle is isosceles.

If both statements are given, then you can find the area without making such assumptions.

 

[Acecustis]

How do you know that AD bisects the triangle? Are you assuming from the picture that the triangle appears to be isosceles?

In general an altitude of a triangle (AD) is not also a perpendicular bisector; this is only true if the triangle is isosceles.

If both statements are given, then you can find the area without making such assumptions.

Got it. So despite AD forming a 90 degree angle with BC I still shouldn't assume that it is a bisector unless stated or if stated that the triangle is isosceles.

Thanks.

 

[Dr.Peterson]

Right. In fact, you not only shouldn't assume; you can't assume! Only use what is stated in words or clearly indicated by the diagram (such as that D lies between B and C).

I imagine the fact that the picture made it look isosceles (and made D look like the midpoint) led you to subconsciously think that was true, or at least to forget that the concepts of altitude and perpendicular bisector are distinct. "Perpendicular" doesn't mean "bisector".

I could say this is a defect in the problem, as they shouldn't use a picture that misleads, if they are trying to communicate clearly; but it may be an intentional lesson in how not to use a diagram in a proof problem. I discussed several issues related to this in my blog last year, here.

 

[Acecustis]

Right. In fact, you not only shouldn't assume; you can't assume! Only use what is stated in words or clearly indicated by the diagram (such as that D lies between B and C).

I imagine the fact that the picture made it look isosceles (and made D look like the midpoint) led you to subconsciously think that was true, or at least to forget that the concepts of altitude and perpendicular bisector are distinct. "Perpendicular" doesn't mean "bisector".

I could say this is a defect in the problem, as they shouldn't use a picture that misleads, if they are trying to communicate clearly; but it may be an intentional lesson in how not to use a diagram in a proof problem. I discussed several issues related to this in my blog last year, here.

Definitely intentional. The GMAT likes to throw tricks in there to fool test takers.