joylive wrote:
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 triangleCase 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 triangleSince we cannot answer the
target question with certainty, statement 1 is NOT SUFFICIENT
Statement 2: AC is the greatest side of triangle ACDKEY 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 triangleSince we can answer the
target question with certainty, statement 2 is SUFFICIENT
Answer: B
Cheers,
Brent