In the xy-plane, triangle ABC has vertices \(\mathrm{A(-12, 4)}\), \(\mathrm{B(13, 4)}\), and \(\mathrm{C(7, 32)}\). What is the area, in square...

1. TRANSLATE the coordinate information

• Given vertices:
  • \(\mathrm{A(-12, 4)}\)
  • \(\mathrm{B(13, 4)}\)
  • \(\mathrm{C(7, 32)}\)

• What this tells us: We have three points that form a triangle on the coordinate plane.

2. INFER the optimal strategy

• Key insight: Points A and B both have \(\mathrm{y = 4}\)
• This means segment AB is horizontal, making it perfect as our base
• Using a horizontal base simplifies finding the height (no complex perpendicular calculations needed)

3. TRANSLATE coordinates into measurements

• Base length AB = horizontal distance = \(\mathrm{13 - (-12) = 25}\)
• Height = vertical distance from \(\mathrm{C(7, 32)}\) to line AB (which lies on \(\mathrm{y = 4}\))
• Height = \(\mathrm{|32 - 4| = 28}\)

4. SIMPLIFY using the area formula

• \(\mathrm{Area = \frac{1}{2} \times base \times height}\)
• \(\mathrm{Area = \frac{1}{2} \times 25 \times 28}\)
• \(\mathrm{= \frac{1}{2} \times 700}\)
• \(\mathrm{= 350}\)

Answer: B (350)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students miss that A and B have the same y-coordinate, so they attempt to use the complex coordinate area formula: \(\mathrm{Area = \frac{1}{2}|x_1(y_2-y_3) + x_2(y_3-y_1) + x_3(y_1-y_2)|}\). While this formula works, it's much more prone to arithmetic errors and sign mistakes. Students often make calculation errors in the multiple steps, potentially leading them to select Choice A (175) or Choice C (364).

Second Most Common Error:

Poor TRANSLATE execution: Students correctly identify AB as the horizontal base but calculate the base length incorrectly as \(\mathrm{|13 + 12| = 1}\) instead of \(\mathrm{|13 - (-12)| = 25}\), or miscalculate the height. These errors in basic distance calculations often lead to Choice D (400) or other incorrect values.

The Bottom Line:

The key insight that makes this problem manageable is recognizing the horizontal base formed by points A and B. Students who miss this geometric simplification end up using more complex formulas and making more arithmetic errors.