**GRE Coordinate Geometry: Slope, Distance, Midpoint, Line Equations**

GRE Coordinate Geometry: Slope, Distance, Midpoint, Line Equations

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What This Is

Coordinate geometry appears in ~5–7% of GRE Quant questions—typically 2–3 problems per test. These questions test your ability to: - Calculate slopes, distances, midpoints, and line equations from points or graphs.
- Interpret parallel/perpendicular lines and intercepts.
- Solve word problems involving geometric figures in the coordinate plane.

Mastering this topic boosts your score by 2–4 points because: ✅ It’s formula-light (only 4–5 key equations to memorize).
✅ It’s predictable (ETS recycles the same question structures).
✅ It’s fast (most problems take <90 seconds if you follow the strategy below).

Real GRE-Style Example:
In the xy-plane, line k passes through the points (–2, 3) and (4, –1). Which of the following is an equation of a line perpendicular to k? (A) y = –x + 5 (B) y = x – 2 (C) y = 2x + 1 (D) y = –½x + 3 (E) y = ½x – 4

(Answer: D. Perpendicular lines have slopes that are negative reciprocals. Slope of k = –⅔, so perpendicular slope = ½. Only D has slope ½.)

Key Concepts & Techniques

1. Slope Formula (m = (y₂ – y₁)/(x₂ – x₁))
2. When to use: When you have two points and need the slope of the line through them.

3. Pro tip: If x₂ = x₁, the line is vertical (slope = undefined). If y₂ = y₁, the line is horizontal (slope = 0).

4. Distance Formula (d = √[(x₂ – x₁)² + (y₂ – y₁)²])

5. When to use: To find the length of a segment between two points.

6. Shortcut: If the points are horizontal/vertical, just subtract the differing coordinates (e.g., distance between (3, 5) and (3, 9) = |9 – 5| = 4).

7. Midpoint Formula (M = ((x₁ + x₂)/2, (y₁ + y₂)/2))

8. When to use: To find the center point of a segment (e.g., for circles, triangles, or symmetry problems).

9. Slope-Intercept Form (y = mx + b)

10. When to use: To write the equation of a line when you know the slope (m) and y-intercept (b).

11. GRE trap: If the line is vertical, it’s x = a (not y = mx + b).

12. Point-Slope Form (y – y₁ = m(x – x₁))

13. When to use: When you have one point and a slope (e.g., "a line passes through (2, –3) with slope 4").

14. Convert to slope-intercept by solving for y.

15. Parallel & Perpendicular Lines

16. Parallel: Same slope (m₁ = m₂).
17. Perpendicular: Slopes are negative reciprocals (m₁ × m₂ = –1).

18. When to use: For questions asking about relationships between lines (e.g., "Which line is perpendicular to y = 3x + 2?").

19. Intercepts (x-intercept = (a, 0), y-intercept = (0, b))

20. When to use: To graph a line quickly or solve for missing values (e.g., "A line has x-intercept 4 and y-intercept –2. What’s its equation?").

21. Equation from intercepts: x/a + y/b = 1.

22. Reflections Over Axes

23. Reflect over x-axis: (x, y) → (x, –y).
24. Reflect over y-axis: (x, y) → (–x, y).
25. When to use: For symmetry problems (e.g., "Point (3, –5) is reflected over the y-axis. What’s the new point?").

Step-by-Step Strategy

Follow these steps for EVERY coordinate geometry question:

1. Identify what’s given and what’s asked.
2. Points? Slope? Intercepts? Parallel/perpendicular?

3. Example: "Line k passes through (–1, 2) and (3, –4). What’s the equation of a line perpendicular to k?"

4. Calculate the slope (if needed).

5. Use m = (y₂ – y₁)/(x₂ – x₁).

6. If the question involves parallel/perpendicular lines, find the slope first.

7. Use the appropriate formula.

8. Need distance? → Distance formula.
9. Need midpoint? → Midpoint formula.

10. Need line equation? → Point-slope or slope-intercept form.

11. Simplify and match answer choices.

12. Convert to slope-intercept form (y = mx + b) for easy comparison.

13. Eliminate wrong answers by checking slopes/intercepts.

14. Check for traps.

15. Vertical/horizontal lines? Negative reciprocals? Intercepts at zero?

Fully Worked GRE-Style Example

Question:
In the xy-plane, line m passes through the points (–3, 5) and (1, –1). Line n is perpendicular to line m and passes through the point (2, 4). What is the y-intercept of line n?

Step 1: Identify given and asked.
- Given: Points (–3, 5) and (1, –1) on line m; line n is perpendicular to m and passes through (2, 4).
- Asked: y-intercept of line n.

Step 2: Calculate slope of line m.
m = (y₂ – y₁)/(x₂ – x₁) = (–1 – 5)/(1 – (–3)) = –6/4 = –3/2.

Step 3: Find slope of line n (perpendicular).
Perpendicular slope = negative reciprocal of –3/2 = 2/3.

Step 4: Write equation of line n using point-slope form.
Point: (2, 4), slope = 2/3.
y – 4 = (2/3)(x – 2).

Step 5: Convert to slope-intercept form (y = mx + b).
y – 4 = (2/3)x – 4/3 y = (2/3)x – 4/3 + 4 y = (2/3)x + (–4/3 + 12/3) y = (2/3)x + 8/3.

Step 6: Identify y-intercept.
y-intercept = 8/3.

Answer: 8/3.

Common Mistakes

1. Mistake: Forgetting that vertical lines have undefined slope.
2. Why it happens: Students assume all lines have a slope.

3. Correct approach: If x-coordinates are equal (e.g., (2, 3) and (2, 7)), the line is x = 2 (vertical).

4. Mistake: Mixing up parallel/perpendicular slopes.

5. Why it happens: Students confuse "same slope" (parallel) with "negative reciprocal" (perpendicular).

6. Correct approach: Write down the rule: Parallel = same slope. Perpendicular = negative reciprocal.

7. Mistake: Misapplying the distance formula for horizontal/vertical lines.

8. Why it happens: Students overcomplicate simple cases.

9. Correct approach: If points share an x or y coordinate, just subtract (e.g., distance between (5, 2) and (5, 8) = 6).

10. Mistake: Forgetting to simplify line equations.

11. Why it happens: Students leave answers in point-slope form (e.g., y – 2 = 3(x + 1)) instead of slope-intercept.

12. Correct approach: Always convert to y = mx + b for easy comparison with answer choices.

13. Mistake: Ignoring intercepts when graphing.

14. Why it happens: Students focus only on slope.
15. Correct approach: For line equations, find both intercepts to sketch the line quickly.

GRE Traps & Timing

1. Trap: "Which of the following is parallel/perpendicular?"
2. How to spot: The question gives a line and asks for another line with a specific relationship.

3. How to avoid: Always calculate the slope first, then apply the parallel/perpendicular rule.

4. Trap: Hidden vertical/horizontal lines.

5. How to spot: Points with the same x or y coordinate (e.g., (4, –2) and (4, 5)).

6. How to avoid: Check if x₁ = x₂ (vertical) or y₁ = y₂ (horizontal) before calculating slope.

7. Trap: Midpoint vs. distance confusion.

8. How to spot: The question asks for a "center" or "halfway point" (midpoint) but students calculate distance.

9. How to avoid: Read carefully—midpoint = average of coordinates, distance = √[(x₂ – x₁)² + (y₂ – y₁)²].

10. Timing:

11. Easy question (e.g., slope between two points): <60 seconds.
12. Medium question (e.g., perpendicular line equation): 60–90 seconds.
13. Hard question (e.g., word problem with multiple steps): 90–120 seconds.

Quick Practice

1. What is the slope of the line passing through (–4, 7) and (2, –5)?
2. Answer: –2.

3. Solution: m = (–5 – 7)/(2 – (–4)) = –12/6 = –2.

4. Line p has equation y = 3x – 4. Which of the following lines is perpendicular to p?
   (A) y = –3x + 1
   (B) y = (1/3)x – 2
   (C) y = –(1/3)x + 5
   (D) y = 3x + 2

5. Answer: C.
6. Solution: Perpendicular slope = –1/3 (negative reciprocal of 3). Only C has slope –1/3.

Last-Minute Cram Sheet

1. Slope formula: m = (y₂ – y₁)/(x₂ – x₁).
2. Distance formula: d = √[(x₂ – x₁)² + (y₂ – y₁)²].
3. Midpoint formula: M = ((x₁ + x₂)/2, (y₁ + y₂)/2).
4. Slope-intercept form: y = mx + b (m = slope, b = y-intercept).
5. Point-slope form: y – y₁ = m(x – x₁).
6. Parallel lines: Same slope (m₁ = m₂).
7. Perpendicular lines: Slopes are negative reciprocals (m₁ × m₂ = –1).
8. Vertical line: x = a (slope = undefined).
9. Horizontal line: y = b (slope = 0).
10. ⚠️ Trap: If a line passes through the origin, its y-intercept is 0 (not "no intercept").

Memorize these, and you’ll solve 90% of GRE coordinate geometry questions in under 90 seconds.