Coordinate Geometry

1Summary 重點摘要

Learn two point distance, midpoint, slope, line equation.

2Vocabulary cards Vocabulary

3Step by step Steps

1. Step 1: The distance formula — Pythagoras in coordinatesThe distance between A(x₁, y₁) and B(x₂, y₂) is d = √[(x₂-x₁)² + (y₂-y₁)²]. It is Pythagoras' theorem in disguise: the horizontal and vertical gaps between the points form the two legs of a right triangle, and the distance is the hypotenuse. Mark the coordinates first, then substitute term by term — subtract inside each bracket before squaring. The two subtractions must keep the same order of points; forgetting to square a negative, or forgetting the final square root, are the common mark-losers.
2. Step 2: The midpoint formula — average each coordinateThe midpoint of AB is ((x₁+x₂)/2, (y₁+y₂)/2): average the x-coordinates and average the y-coordinates, because the midpoint lies exactly halfway in both directions. The formula appears often in proofs: a quadrilateral whose diagonals bisect each other is a parallelogram, shown by comparing the midpoints of the two diagonals. Do not average only one coordinate, and never confuse the midpoint formula with the distance formula; keep x₁ paired with y₁ and x₂ with y₂ — crossed pairing is a frequent slip.
3. Step 3: The slope formula — rise over runThe slope of the line through two points is m = (y₂-y₁)/(x₂-x₁): the vertical change over the horizontal change. Slope measures the tilt of a line: m > 0 rises, m < 0 falls, m = 0 is horizontal, and a zero denominator means the line is vertical with undefined slope. The same point must lead in the numerator and the denominator; writing (y₂-y₁)/(x₁-x₂) flips the sign. Confirm first that the two x-coordinates differ, otherwise no slope exists.
4. Step 4: The point-slope form — the key to any equationGiven the slope m and a point (x₁, y₁) on the line, the equation is y - y₁ = m(x - x₁), because every point on the line forms the same slope m with the fixed point. Simplify the result to y = mx + c so the slope and the y-intercept can be read off easily. The fixed point goes into (x₁, y₁); the letters x and y remain as variables in the equation. Watch the signs while simplifying, especially subtracting a negative, which turns into addition.
5. Step 5: Parallel and perpendicular — two slope relationsTwo lines are parallel exactly when their slopes are equal, m₁ = m₂, since lines pointing the same way tilt equally. When both slopes exist, two lines are perpendicular exactly when m₁ × m₂ = -1, that is, one slope is the negative reciprocal of the other: the line perpendicular to a slope of 2 has slope -12. Use these facts to find the new slope first, then apply the point-slope form. Beware: flipping the fraction without changing the sign, and vertical lines have no slope at all — a vertical and a horizontal line are perpendicular by picture, not by formula.
6. Step 6: Sketch to verify and combine the toolsAfter computing, plot the key points for a rough check: is the distance sensible, is the midpoint centred on the segment, does the line pass through the given point? Multi-part questions chain the formulas: find slopes to classify the shape, use the distance formula to prove two sides equal, then use midpoints to prove the diagonals bisect each other. Working without any sketch hides substituted errors; a rough drawing need not be accurate but catches obvious mistakes at almost no cost.

4Diagrams Diagrams

5Worked example Example

6Common mistakes Mistakes

7Quick check Quick check

8Recap Recap

• Distance: √[(x diff)² + (y diff)²]
• Midpoint: average x, average y
• Slope: (y diff)/(x diff)
• Parallel: same slope
• Perpendicular: slope multiply = -1