If ABD is a triangle, is triangle ABC a right triangle

IMO answer is B.

The question is basically asking whether any of the angles in triangle ABC is 90.
1) Having told that ACD is a right triangle, doesn't confirms whether any of the angle in ABC is 90. The statement only lets us know that there is a 90 degree angle in ACD, but which one, we dont know yet.

2) This statement tells that angle D is the greatest so in other words it means that angles DCA or DAC cannot be 90. Moreover, since angle C will always be less than 90 degrees hence its supplement will always be greater than 90.
Since one angle in ABC is already greater than 90, therefore either of the other two angles cannot be equal to 90.
Sufficient.

+1B

A student asked me to comment on this question. So.....


Target question: Is triangle ABC a right triangle?

Statement 1: ACD is a right triangle.
This means ONE of the 3 angles in ∆ACD is 90°
So, let's examine 2 possible cases:
Case a: ∠DCA = 90°. This means ∠ACB = 90°, in which case, the answer to the target question is YES, ∆ABC IS a right triangle
Case b: ∠ADC = 90°. This means ∠DCA < 90°, which means ∠ACB > 90°. If ∠ACB > 90°, then no other angle in ∆ABC can be 90°. In which case, the answer to the target question is NO, ∆ABC is NOT a right triangle
Since we cannot answer the target question with certainty, statement 1 is NOT SUFFICIENT

Statement 2: AC is the greatest side of triangle ACD
KEY CONCEPT: The angle opposite the longest side will be the largest angle.
This means ∠ADC is the largest angle



If ∠ADC is the largest angle, then the other two angles in ∆ACD must be less than 90°

ASIDE: How can we conclude that the other two angles in ∆ACD must be less than 90°?
Well, let's say that one angle (say ∠ADC) is greater than 90°. Since ∠ADC is the largest angle, this would mean that ∠ADC is also greater than 90°
At this point, we have TWO angles in a triangle that are BOTH greater than 90°. This would mean that the sum of the triangle's angles would be greater than 180°, which is IMPOSSIBLE.
So, it MUST be the case that the other two angles in ∆ACD must be less than 90°


Since ∠DCA is LESS than 90°, we can see that ∠ACB is GREATER THAN than 90° (since those two angles must add to 180°)



Finally, ∠ACB is GREATER THAN than 90°, the two other angles in ∆ABC must be LESS THAN 90° (since we can't have TWO angles in a triangle that are both greater than 90°)

So, ∆ABC had 1 angle greater than 90° and two angles less than 90°, we know that ∆ABC cannot be a RIGHT triangle.

The answer to the target question is NO, ∆ABC is definitely NOT a right triangle

Since we can answer the target question with certainty, statement 2 is SUFFICIENT

Answer: B

Cheers,
Brent

The answer would be A.

Statement 1 is sufficient because:

Case 1: Angle ACD is a right angle then angle ACB has to be 180 - 90 = 90 degrees => ABC a right triangle
Case 2: Angle CAD is a right angle then both angle ADC and ACD are less than 90 degrees. Hence Angle ACB is greater than 90 degrees hence ABC is not a right triangle
Case 3: Angle ADC is a right angle then by similar logic ABC can not be a right triangle


Statement 2 is insufficient because:

if AC is the greatest side of triangle ACD then this only tells us that the angle ADC is the largest angle in the said triangle. Note that this angle CAN be 90 degree or less than or greater than 90 degree. Barring the first case we cannot conclusive say about the triangle ABC.

I can see that you got 2 different answers for Statement I, so is this sufficient?

Hi,

Correct me if I'm wrong.

1. One of the unspoken rules in DS Questions is " The two options in the questions cannot contradict each other"
2. In your explanation for option 2, you have interpreted greatest SIDE for greatest ANGLE which are not the same.

I doubt the diagram given in the Q is imagined by the person who posted the Q initially, because clearly, in the given picture AC is not the greatest side of the triangle ACD.

In my opinion, the solution would be,

Q statement gives us the detail, that ABD and ABC are triangles.

option 1 states :

ACD is a right triangle, which means one of the three angles is 90'.INSUFFICIENT.

option 2 states:

AC is the greatest side of triangle ACD. with this we can draw a triangle ACD with AC as the hypotenuse and angle ADC = 90' ( refer diagram 1). INSUFFICIENT.

Now,
with the given data together in both the options ABC may or may not be a right triangle.( refer diagram 2 and 3). In case of 2 - yes and in case of 3 - No.
So, the answer is E. insufficient data.

plz share your inputs.

thank you.

I guess the answer has to be B

Statement 1/
Sum of the angles in a triangle = 180* & sum of ACD and ACB should equal 180*
If ACD is 90* then ACB should also be 90* - ABC a right triangle
but if DAC = 90* then other 2 angles of ACD must be less than 90* say ADC = 60 and ACD = 30 then ACB =150 and other 2 angles can be 10 and 20* - ABC is NOT a right angled triangle.

Statement 2/
IF AC is greatest side the angle at D must be greatest among the 3 angles of the triangle.
then the other 2 angles cannot be 90* or more if that's the case then ABC CANNOT BE right angled triangle. B is sufficient.

Cheers

Coz I don't have the OA, if I had it, I would have already have posted it.Got these Q's somewhere and trying to get the answer myself.

U shud have mentioned earlier :D
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in fact joylive has done so. :D

Cheers

Hi
can anybody help with this? I also think answer should be E based on imk's solution.
Thanks

The second diagram in that post is not valid. The figure in the question says that point C is on side DB, which is violated there.
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Hi thanks,
So B is the answer , I guess , can you provide the explanation if possible , or point to the best solution.
Thanks

Check here: if-abd-is-a-triangle-is-triangle-abc-a-right-triangle-143775.html#p1152300 or here: if-abd-is-a-triangle-is-triangle-abc-a-right-triangle-143775.html#p1152309 Please ask if anything is unclear there.
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The answer to the question is "C". OP must have been using a study guide that ripped it from the original source and messed up the answer key.

The gist of the answer is that we are making the assumption that Angle ACD is the right angle and that line AC runs perpendicular to BD. However, taking into account both statements, we know that ABC is a right angle and AC is the hypotenuse.

Another thing, the book the question was taken from says, "The dimensions in the figure may be different from what they appear to be" underlined directly under the figure.

I've read Nova's solution and it does not make sense. If AC is the greatest side of triangle ACD, then none of the angles of triangle AVC can be right.
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I'll be honest, I half-ass read the solution and think that I got where they're coming from. It's the way I understand it that makes Choice C the best answer.

Angle D is the right-angle since AC is the longest segment of ADC. Consequently, if Angle D is the right, AC is the hypotenuse, it would make Angle C an obtuse angle greater than 90 and subsequently make angle A and B less than 90. Hence... ABC is not a right triangle.

I've probably done 300+ nova problems and one them is that you cannot take their drawings for granted.... at all. They push the "drawing may not be to scale" to the limit.

You are talking about the case when we combine the statements. I'm talking about the second statement alone.

What I'm saying is that if AC is the greatest side of triangle ACD, then none of the angles of triangle ABC can be right.

I'm not a big fan of Nova's question... Not a good source from my opinion.
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pls can someone further explain the answer B? just dont get it! Think it should be A

The diagram is way too misleading . I agree that in GMAT one can not assume anything in a figure but this is taking it to another level