Grade: Grade 10 Subject: Mathematics Unit: Coordinate Geometry Lesson: 5 of 6 SAT: Geometry+Trigonometry ACT: Math

Common Mistakes

📖 Learn

Understanding common errors in coordinate geometry helps you avoid them on tests and assignments. This lesson identifies frequent mistakes and shows you how to prevent them.

Why Study Mistakes?

  • Mistakes reveal gaps in understanding
  • Recognizing error patterns helps you self-check
  • SAT and ACT often include answer choices that match common errors
  • Awareness prevents careless errors under time pressure

Categories of Common Mistakes

  1. Sign errors - Mishandling negative numbers
  2. Formula confusion - Mixing up distance, midpoint, and slope
  3. Order errors - Swapping x and y values
  4. Calculation errors - Arithmetic mistakes under the radical
  5. Interpretation errors - Misunderstanding what the answer represents

💡 Examples

Study these common mistakes and their corrections.

Mistake 1: Sign Errors with Negative Coordinates

Problem: Find the distance between A(3, -2) and B(-1, 4).

Wrong approach:

d = sqrt[(3 - (-1))^2 + (-2 - 4)^2] - but student writes (-2 + 4) = 2

Correct approach:

d = sqrt[(3 - (-1))^2 + (-2 - 4)^2] = sqrt[16 + 36] = sqrt[52]

Key insight: When subtracting, (-2) - 4 = -6, not 2. Always be careful with double negatives.

Mistake 2: Forgetting to Square Both Terms

Problem: Find the distance between (0, 0) and (3, 4).

Wrong approach:

d = sqrt[3 + 4] = sqrt[7] (forgot to square)

Correct approach:

d = sqrt[3^2 + 4^2] = sqrt[9 + 16] = sqrt[25] = 5

Key insight: The distance formula requires squaring both differences before adding.

Mistake 3: Swapping Slope Formula

Problem: Find the slope through (2, 5) and (6, 9).

Wrong approach:

m = (x2 - x1)/(y2 - y1) = (6-2)/(9-5) = 4/4 = 1

Correct approach:

m = (y2 - y1)/(x2 - x1) = (9-5)/(6-2) = 4/4 = 1

Key insight: This happened to give the same answer, but slope is "rise over run" - y-change over x-change. Remember: y comes first!

Mistake 4: Midpoint vs. Distance Confusion

Problem: Find the midpoint of segment AB where A(2, 8) and B(6, 4).

Wrong approach:

Using distance formula: sqrt[(6-2)^2 + (4-8)^2] = sqrt[32]

Correct approach:

M = ((2+6)/2, (8+4)/2) = (4, 6)

Key insight: Midpoint uses addition and division by 2. Distance uses subtraction, squaring, and square roots.

Mistake 5: Division by Zero in Slope

Problem: Find the slope through (5, 2) and (5, 8).

Wrong approach:

m = (8-2)/(5-5) = 6/0 = 0 (student writes 0 instead of undefined)

Correct approach:

m = 6/0 = undefined (vertical line)

Key insight: Division by zero is undefined. A vertical line has no slope (not zero slope). Zero slope means horizontal.

✏️ Practice

Find and correct the mistake in each problem. Explain what went wrong.

1. Student work: Distance from (-3, 2) to (5, -4) is sqrt[(5+3)^2 + (-4+2)^2] = sqrt[64 + 4] = sqrt[68]. Is this correct?

2. Student work: Midpoint of (4, 10) and (8, 2) is ((4-8)/2, (10-2)/2) = (-2, 4). Find the error.

3. Student work: Slope through (1, 3) and (4, 3) is (4-1)/(3-3) = 3/0 = undefined. Is this correct?

4. Student work: Distance from origin to (5, 12) is 5 + 12 = 17. Find the error.

5. Student work: The slope of a horizontal line y = 5 is undefined. Is this correct?

6. Student work: Midpoint of (-2, 6) and (4, -8) is ((-2+4)/2, (6+(-8))/2) = (3, -1). Find the error.

7. Student work: To find if (3, 7) lies on line through (0, 1) and (2, 5), check slope: (7-1)/(3-0) = 6/3 = 2 and (5-1)/(2-0) = 4/2 = 2. Student concludes point is on the line. Is this correct?

8. Student work: Distance between (a, b) and (b, a) is always 0 because they have the same numbers. Is this correct?

9. Student work: If the midpoint is (3, 5) and one endpoint is (1, 2), the other endpoint is (3-1, 5-2) = (2, 3). Find the error.

10. Student work: Parallel lines have slopes that multiply to -1. Is this statement correct?

✅ Check Your Understanding

Review the corrections for each practice problem.

1. This is actually correct! But note: (-4-2) = -6, and (-6)^2 = 36, not 4. The student got lucky because (-4+2) = -2 and (-2)^2 = 4 happens to be different. Wait - check again: (-4) - 2 = -6, (-6)^2 = 36. Answer should be sqrt[64 + 36] = 10. Student made a sign error.

2. Midpoint uses addition, not subtraction. Correct: ((4+8)/2, (10+2)/2) = (6, 6)

3. No - student swapped x and y. The slope should be (3-3)/(4-1) = 0/3 = 0. Horizontal line has slope 0.

4. Must use distance formula: sqrt[25 + 144] = sqrt[169] = 13, not simple addition.

5. No - horizontal lines have slope 0. Vertical lines (like x = 5) have undefined slope.

6. Error in first coordinate: ((-2)+4)/2 = 2/2 = 1, not 3. Correct midpoint: (1, -1)

7. Correct! The point lies on the line because slopes are equal.

8. No - unless a = b. Distance = sqrt[(b-a)^2 + (a-b)^2] = sqrt[2(a-b)^2] = |a-b|sqrt[2]

9. Must double the distance from midpoint. If M = (3,5) and A = (1,2), then B = (2(3)-1, 2(5)-2) = (5, 8)

10. No - that describes perpendicular lines. Parallel lines have equal slopes.

🚀 Next Steps

  • Create a personal checklist of mistakes to watch for
  • Practice double-checking your work using these error categories
  • On timed tests, budget time to verify your calculations
  • When you are ready, take the Unit Quiz to test your mastery