Hi, there! I'm happy to help with this.
We are given Triangle ABD with a right angle at B. Then, C is placed on BD, and line AC is constructed.
The question asks: is triangle ABC isosceles. We already know that triangle ABC has one right angle in it at B. A triangle can't have two right angles, so the only way triangle ABC can be isosceles is if AB = BC and, with that, angle BAC = angle ACB, with both of them having a value of 45 degrees. Any information along those lines would help us establish that triangle ABC is isosceles. Statement #1
: Angle ACB is twice as large as angle ADC
We have no idea what the angles are in triangle ADC, so knowing that angle ACB is twice as large as angle ADC tells us nothing about the measure of ACB, nor anything about the relationship of angle ACB and angle CAB. This statement is entirely insufficient. Statement #2
: Angle ACB is twice as large as angle CAB
This tells us a crucial piece of information. Triangle ABC would be isosceles if and only if angles ACB & CAB are congruent. Statement #2 tells us angle ACB is twice as large as angle CAB --- therefore, they are not
congruent, therefore triangle ABC is definitely not
isosceles. Statement #2 is sufficient for answering the question.
Answer = B
Here's another geometry question, just for practice. http://gmat.magoosh.com/questions/1009
Does all that make sense? Please let me know if you have any questions.
Magoosh Test Prep