Triangle ABC is inscribed in a rectangle ABEF...

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Triangle ABC is inscribed in a rectangle ABEF forming two right triangles: AFC and BEC. Is triangle ABC an equilateral triangle?
• a)
Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked.
• b)
Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient to answer the question asked.
• c)
BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient to answer the question asked.
• d)
• e)
Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data specific to the problem are needed.

Steps 1 & 2: Understand Question and Draw Inferences
Given:
The given information corresponds to the following figure:
To find: Is triangle ABC equilateral?

Step 3: Analyze Statement 1 independently
• Let’s drop a perpendicular CD on side AB.
• But the question is, is D the mid-point of AB?
We do not know.
Therefore, Statement 1 is not sufficient to answer the question.

Step 4: Analyze Statement 2 independently
• Point C is the midpoint of EF
• Let’s drop a perpendicular CD on side AB
• CD is parallel to and equal to BE.
• Since C is the mid-point of EF, D will be the mid-point of AB.
• But, we don’t know the magnitude of either AB or CD or AC. So, we cannot find the angles of the triangle.
Not sufficient.

Step 5: Analyze Both Statements Together (if needed)
• From Statement 1: If D is the mid-point of AB, then triangle ABC is an equilateral triangle
• From Statement 2: D is the mid-point of AB

Thus, the two statements together are sufficient to answer the question.

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