ACT Prep: Coordinate Geometry (Lines, Circles, Distance Formula, Midpoint)

ACT – Coordinate Geometry (Lines, Circles, Distance Formula, Midpoint)

ACT Coordinate Geometry Study Guide

Lines, Circles, Distance Formula, Midpoint

What This Is

Coordinate geometry appears on every ACT Math test (typically 4–6 questions). You’ll work with points, lines, slopes, distances, midpoints, and circles on the xy-plane. Real-world example: A city planner uses coordinate geometry to determine the shortest route between two subway stations (distance formula) or the center of a circular park (circle equation). A typical ACT question might ask: "What is the equation of the line passing through (2, 3) and perpendicular to y = –4x + 1?"

Key Terms & Rules

• Coordinate Plane (xy-plane): A grid with an x-axis (horizontal) and y-axis (vertical) where points are plotted as (x, y).
• Slope (m): The steepness of a line; m = (y₂ – y₁)/(x₂ – x₁). Positive slope = rises left to right; negative slope = falls left to right.
• Slope-Intercept Form: y = mx + b; m = slope, b = y-intercept (where the line crosses the y-axis).
• Point-Slope Form: y – y₁ = m(x – x₁); used when you know a point and the slope.
• Standard Form of a Line: Ax + By = C; A, B, and C are integers (no fractions/decimals).
• Parallel Lines: Have the same slope (e.g., y = 2x + 1 and y = 2x – 5).
• Perpendicular Lines: Slopes are negative reciprocals (e.g., y = 3x + 2 and y = –1/3x + 4).
• Distance Formula: d = √[(x₂ – x₁)² + (y₂ – y₁)²]; finds the length between two points.
• Midpoint Formula: M = ((x₁ + x₂)/2, (y₁ + y₂)/2); finds the center point between two coordinates.
• Equation of a Circle: (x – h)² + (y – k)² = r²; (h, k) = center, r = radius.
• Vertical Lines: x = a (undefined slope; e.g., x = 3).
• Horizontal Lines: y = b (slope = 0; e.g., y = –2).

Step-by-Step / Process Flow

How to Solve a Line Equation Question

1. Identify what’s given: A point, slope, or another line (parallel/perpendicular).
2. Find the slope (if needed):
3. If given two points, use m = (y₂ – y₁)/(x₂ – x₁).
4. If parallel, use the same slope; if perpendicular, use the negative reciprocal.
5. Plug into point-slope or slope-intercept form:
6. Use y – y₁ = m(x – x₁) if you have a point and slope.
7. Convert to y = mx + b if needed.
8. Check for standard form: If the question asks for Ax + By = C, rearrange and eliminate fractions.
9. Verify: Plug the given point back into your equation to ensure it works.

How to Solve a Distance/Midpoint Question

1. Label the points: Assign (x₁, y₁) and (x₂, y₂).
2. Apply the formula:
3. Distance: d = √[(x₂ – x₁)² + (y₂ – y₁)²]
4. Midpoint: M = ((x₁ + x₂)/2, (y₁ + y₂)/2)
5. Simplify: For distance, combine under the square root first; for midpoint, simplify each coordinate.
6. Check units: If the question asks for distance in "units," don’t add extra labels.

How to Solve a Circle Equation Question

1. Identify the center and radius: Rewrite the equation in standard form (x – h)² + (y – k)² = r².
2. Complete the square (if needed): If the equation is x² + y² + Dx + Ey + F = 0, group x and y terms, complete the square, and move constants to the other side.
3. Answer the question: If asked for the radius, take the square root of r²; if asked for the center, read (h, k) directly.

Common Mistakes

• Mistake: Mixing up the order of points in the distance/midpoint formulas.
  Correction: Always subtract x₂ – x₁ and y₂ – y₁ (not x₁ – x₂). For midpoint, add first, then divide.

• Mistake: Forgetting to take the negative reciprocal for perpendicular slopes.
  Correction: If the original slope is m, the perpendicular slope is –1/m. Example: m = 2 → perpendicular slope = –1/2.

• Mistake: Misidentifying the center of a circle from its equation.
  Correction: The center is (h, k), not (–h, –k). Example: (x – 3)² + (y + 2)² = 25 → center = (3, –2).

• Mistake: Assuming all lines with the same y-intercept are parallel.
  Correction: Parallel lines must have the same slope, not just the same y-intercept.

• Mistake: Forgetting to square the radius in the circle equation.
  Correction: If the radius is r, the equation uses r². Example: Radius = 5 → equation uses 25.

Exam Insights

• Most-tested concept: Slope (parallel/perpendicular lines, finding slope from two points).
• Tricky distractor: ACT often gives answer choices with slope signs flipped (e.g., –3 instead of 3) or reciprocals (e.g., 1/2 instead of 2).
• Circle questions: Usually test completing the square or identifying the center/radius from the equation.
• Distance/midpoint: Often paired with word problems (e.g., "A drone flies from (1, 2) to (4, 6). How far did it travel?").
• Calculator tip: Use the distance formula on your calculator to avoid arithmetic errors (e.g., type √((4–1)² + (6–2)²)).

Quick Check Questions

1. What is the slope of the line perpendicular to y = –3/4x + 5?
   A) 3/4
   B) –4/3
   C) 4/3
   D) –3/4
   Answer: C) 4/3. Perpendicular slopes are negative reciprocals (–3/4 → 4/3).

2. Find the distance between (–1, 5) and (3, –2).
   A) 5
   B) √65
   C) √53
   D) 7
   Answer: B) √65. Distance = √[(3 – (–1))² + (–2 – 5)²] = √(16 + 49) = √65.

3. What is the center of the circle with equation (x + 2)² + (y – 7)² = 16?
   A) (2, –7)
   B) (–2, 7)
   C) (2, 7)
   D) (–2, –7)
   Answer: B) (–2, 7). The equation is (x – (–2))² + (y – 7)² = 16, so center = (–2, 7).

Last-Minute Cram Sheet

1. Slope formula: m = (y₂ – y₁)/(x₂ – x₁).
2. Parallel lines: Same slope. Perpendicular lines: Negative reciprocal slope.
3. Distance formula: d = √[(x₂ – x₁)² + (y₂ – y₁)²].
4. Midpoint formula: M = ((x₁ + x₂)/2, (y₁ + y₂)/2).
5. Circle equation: (x – h)² + (y – k)² = r²; center = (h, k), radius = r.
6. ⚠️ Perpendicular slopes: Flip the fraction and change the sign (e.g., 2/3 → –3/2).
7. ⚠️ Vertical lines: x = a (undefined slope). Horizontal lines: y = b (slope = 0).
8. ⚠️ ACT trap: Answer choices often include the reciprocal of the correct slope (e.g., 1/2 instead of 2).
9. Standard form: Ax + By = C (no fractions, A > 0).
10. ⚠️ Circle center: Watch for signs! (x + 3)² → h = –3, not 3.