In the coordinate plane, triangle ABC has vertices at \(\mathrm{A(2, 7)}\), \(\mathrm{B(-1, 3)}\), and \(\mathrm{C(5, 1)}\). The altitude from vertex...

1. TRANSLATE the problem information

• Given information:
  • Triangle ABC with \(\mathrm{A(2, 7)}\), \(\mathrm{B(-1, 3)}\), \(\mathrm{C(5, 1)}\)
  • Altitude from A to side BC
  • Need to find which point lies on line AD (the altitude)

2. INFER the geometric relationship

• Key insight: An altitude is perpendicular to the side it meets
• This means the slope of altitude AD must be the negative reciprocal of the slope of side BC
• Strategy: Find slope of BC, then find perpendicular slope for AD

3. SIMPLIFY to find the slope of side BC

• Using points \(\mathrm{B(-1, 3)}\) and \(\mathrm{C(5, 1)}\):
• Slope of BC = \(\mathrm{\frac{1 - 3}{5 - (-1)}}\)
  = \(\mathrm{\frac{-2}{6}}\)
  = \(\mathrm{-\frac{1}{3}}\)

4. INFER the slope of the altitude AD

• Since AD ⊥ BC, slope of AD = negative reciprocal of slope of BC
• Slope of AD = \(\mathrm{-\frac{1}{(-\frac{1}{3})}}\)
  = \(\mathrm{3}\)

5. SIMPLIFY to find the equation of line AD

• Using point-slope form with \(\mathrm{A(2, 7)}\) and slope 3:
• \(\mathrm{y - 7 = 3(x - 2)}\)
• \(\mathrm{y - 7 = 3x - 6}\)
• \(\mathrm{y = 3x + 1}\)

6. SIMPLIFY by testing each answer choice

• Substitute each point into \(\mathrm{y = 3x + 1}\):
• (A) \(\mathrm{(-1, 1)}\): \(\mathrm{1 ≟ 3(-1) + 1 = -2}\) ✗
• (B) \(\mathrm{(0, 7)}\): \(\mathrm{7 ≟ 3(0) + 1 = 1}\) ✗
• (C) \(\mathrm{(1, 4)}\): \(\mathrm{4 ≟ 3(1) + 1 = 4}\) ✓
• (D) \(\mathrm{(2, 6)}\): \(\mathrm{6 ≟ 3(2) + 1 = 7}\) ✗

Answer: C (1, 4)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Not recognizing that perpendicular lines have negative reciprocal slopes

Students often know that the altitude is perpendicular, but incorrectly assume this means using the same slope as BC. They might calculate slope of BC as \(\mathrm{-\frac{1}{3}}\), then incorrectly use \(\mathrm{-\frac{1}{3}}\) as the slope of AD instead of 3. This fundamental misunderstanding of perpendicular relationships in coordinate geometry leads to a completely wrong equation for the altitude, causing confusion when testing answer choices and leading to guessing.

Second Most Common Error:

Poor SIMPLIFY execution: Making arithmetic errors when calculating the slope of BC or the negative reciprocal

Students might correctly understand the perpendicular relationship but make sign errors or fraction arithmetic mistakes. For example, calculating the slope of BC incorrectly or making errors when finding the negative reciprocal (like getting -3 instead of 3 for the altitude slope). This leads to wrong equations and may cause them to select an incorrect answer choice or become confused and guess.

The Bottom Line:

This problem tests whether students truly understand the coordinate geometry representation of perpendicular lines, not just the geometric concept of altitude. The key insight is translating 'perpendicular' into 'negative reciprocal slopes.'