In triangle ABC, AB = 4{,680} millimeters (mm) and BC = 4{,680} mm. Which statement is sufficient to prove that...

1. TRANSLATE the problem information

• Given information:
  • Triangle ABC with \(\mathrm{AB = 4{,}680\text{ mm}}\) and \(\mathrm{BC = 4{,}680\text{ mm}}\)
  • Need to find which statement about AC is sufficient to prove the triangle is equilateral
• What this tells us: We already know two sides are equal, and we need to determine what the third side should be.

2. INFER the requirements for an equilateral triangle

• An equilateral triangle must have all three sides equal in length
• Since \(\mathrm{AB = BC = 4{,}680\text{ mm}}\), we need AC to also equal 4,680 mm for the triangle to be equilateral
• Looking at the choices, only choice A gives \(\mathrm{AC = 4{,}680\text{ mm}}\)

3. Verify the logic

• If \(\mathrm{AC = 4{,}680\text{ mm}}\) (choice A), then \(\mathrm{AB = BC = AC = 4{,}680\text{ mm}}\)
• This satisfies the definition of an equilateral triangle
• Choices B, C, and D all give different lengths for AC, which would make the triangle isosceles (two equal sides) but not equilateral

Answer: A

Why Students Usually Falter on This Problem

Most Common Error Path:

Conceptual confusion about triangle types: Students may not clearly understand the difference between isosceles and equilateral triangles, or may not remember that an equilateral triangle requires ALL three sides to be equal.

This leads them to think that having two equal sides (AB = BC) is sufficient to make the triangle equilateral, causing confusion about what the third side needs to be. They might select any of the wrong choices B, C, or D without recognizing that these create isosceles triangles, not equilateral ones.

Second Most Common Error:

Weak INFER reasoning about "sufficient to prove": Students may not understand what it means for a condition to be "sufficient" to prove something. They might focus on whether the triangle could be equilateral rather than what would definitely prove it is equilateral.

This may lead them to select any choice without properly reasoning through the logical requirement, or to get confused and guess randomly.

The Bottom Line:

This problem tests whether students truly understand the precise definition of an equilateral triangle. The key insight is that "equilateral" means ALL sides equal, not just some sides equal.