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805+ Level|   Geometry|      
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If ABD is a triangle, is triangle ABC a right triangle 20 Nov 2019, 07:10
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GMATPrepNow
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Attachment:
Triangle.png
If ABD is a triangle, is triangle ABC a right triangle?

(1) ACD is a right triangle.
(2) AC is the greatest side of triangle ACD.

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 ∆ACD cannot be a RIGHT triangle.

The answer to the target question is YES, ∆ABC IS a right triangle

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

Answer: B

Cheers,
Brent
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Hi Brent do u by any chance mean to say that WE can say for sure that ABC is not a right-angle triangle and hence we can answer this question. Ans = B

I think by mistake you have written yes ABC is a right triangle. Correct me if i am wrong?

I am quoting you below ...


" The answer to the target question is YES, ∆ABC IS a right triangle"
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If ABD is a triangle, is triangle ABC a right triangle 20 Nov 2019, 11:34
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Hi Brent do u by any chance mean to say that WE can say for sure that ABC is not a right-angle triangle and hence we can answer this question. Ans = B

I think by mistake you have written yes ABC is a right triangle. Correct me if i am wrong?

I am quoting you below ...

" The answer to the target question is YES, ∆ABC IS a right triangle"
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Good catch. You're absolutely right.
I meant to say "NO, ∆ABC is NOT a right triangle"
I've edited my response accordingly.

Kudos for you!!!!!!
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If ABD is a triangle, is triangle ABC a right triangle 29 Jun 2020, 09:06
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Marcab
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If ABD is a triangle, is triangle ABC a right triangle?


(1) ACD is a right triangle.
(2) AC is the greatest side of triangle ACD.


OA not given.

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
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You do realize that your explanation is still not fool-proof, right?
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If ABD is a triangle, is triangle ABC a right triangle 29 Jun 2020, 09:12
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Bunuel
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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.

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: https://gmatclub.com/forum/if-abd-is-a-t ... l#p1152300 or here: https://gmatclub.com/forum/if-abd-is-a-t ... l#p1152309 Please ask if anything is unclear there.
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Hi, Bunuel, I just wanted your opinion on this post.
I've been through almost all the threads for this post. And I still believe that (B) alone is not sufficient since angle D may or may not be 90º.
Can you please explain that to me?
Thanks!
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If ABD is a triangle, is triangle ABC a right triangle 11 Sep 2020, 10:56
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BrentGMATPrepNow
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Attachment:
Triangle.png
If ABD is a triangle, is triangle ABC a right triangle?

(1) ACD is a right triangle.
(2) AC is the greatest side of triangle ACD.

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
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Hi BrentGMATPrepNow

It seems like the second angle is not mentioned in the yellow highlighted line or I am not reading it correctly. Kindly correct me if I am wrong. Thanks!

I also had a question - If ∠ADC is the largest angle then can it not be 90, I mean a 45-45-90 case? What makes this angle >90?

Warm Regards,
Pritishd
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If ABD is a triangle, is triangle ABC a right triangle 31 Mar 2024, 21:45
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