Overview
Coordinate geometry connects algebra and geometry through the coordinate plane. The SAT tests the distance formula, midpoint formula, slope calculations, collinearity, transformations (reflections, translations, rotations, dilations), and graphing circles. None of these formulas appear on the SAT reference sheet — all must be memorized. Problems in this area often combine multiple concepts, such as using slope to verify perpendicularity or combining midpoint with distance to solve segment problems.
Key Points
1. The Three Core Formulas (All Must Be Memorized)
Given two points (x₁, y₁) and (x₂, y₂):
Distance: d = √[(x₂ − x₁)² + (y₂ − y₁)²]
Midpoint: M = ((x₁ + x₂)/2 , (y₁ + y₂)/2)
Slope: m = (y₂ − y₁) / (x₂ − x₁)
Memory shortcuts:
- Distance = Pythagorean theorem with coordinate differences as the two legs
- Midpoint = “average the x-values, average the y-values”
- Slope = “rise over run”
Slope relationships:
- Parallel lines: equal slopes (m₁ = m₂)
- Perpendicular lines: slopes are negative reciprocals (m₁ × m₂ = −1)
2. Collinearity
Three points A, B, C are collinear (lie on the same line) if and only if any two of the three slopes between pairs of points are equal.
slope(AB) = slope(BC) → A, B, C collinear
Alternatively, check that all three points satisfy the same linear equation.
3. Partitioning a Segment in a Given Ratio
To find the point P that divides segment AB in ratio m:n from A:
P_x = x₁ + [m/(m+n)] × (x₂ − x₁)
P_y = y₁ + [m/(m+n)] × (y₂ − y₁)
P is located m/(m+n) of the way from A to B.
Example: A = (0, 0), B = (8, 4), ratio 1:3 from A.
- Fraction from A = 1/(1+3) = 1/4
- P = (0 + 1/4 × 8, 0 + 1/4 × 4) = (2, 1)
Special case (m = n): P is the midpoint — same as the midpoint formula.
4. Transformations
| Transformation | Rule |
|---|---|
| Reflection over x-axis | (x, y) → (x, −y) |
| Reflection over y-axis | (x, y) → (−x, y) |
| Reflection over y = x | (x, y) → (y, x) |
| Translation by (a, b) | (x, y) → (x + a, y + b) |
| Rotation 90° CCW about origin | (x, y) → (−y, x) |
| Rotation 180° about origin | (x, y) → (−x, −y) |
| Dilation by factor k (origin) | (x, y) → (kx, ky) |
Dilation not centered at origin: Subtract center coordinates, apply factor k, then add center coordinates back.
5. Graphing Circles in the Coordinate Plane
Standard form: (x − h)² + (y − k)² = r²
- Center: (h, k)
- Radius: r
To graph or analyze a circle given in general form (x² + y² + Dx + Ey + F = 0), complete the square to convert to standard form. See Properties_Circles for the full completing-the-square method.
6. Visual Approach for Distance Problems
For horizontal or vertical segments, distance = difference in coordinates (no formula needed). For diagonal segments, sketch the right triangle with horizontal and vertical legs, then apply the Pythagorean theorem. This visual method is often faster than the formula on the SAT.
Pitfalls and Common Mistakes
Pitfall 1: Forgetting that coordinate formulas are not on the reference sheet Students sometimes expect the distance or midpoint formulas to appear on the SAT formula reference page (as they do in some state tests). They do not appear on the SAT. Fix: Memorize all three formulas before test day. Practice writing them from memory before each timed practice session.
Pitfall 2: Confusing slope direction for perpendicular lines Knowing that perpendicular slopes are “negative reciprocals” but applying it as just “reciprocal” (forgetting the sign flip) gives a line with the right steepness but the wrong direction. Fix: If m₁ = 2/3, the perpendicular slope is −3/2. Always flip both the fraction and the sign.
Pitfall 3: Reversing the ratio direction when partitioning “P divides AB in ratio 1:3 from A” means P is 1/4 of the way from A to B, not 3/4. Reversing this gives the wrong point. Fix: The formula uses m/(m+n) from the first named endpoint. Write out the fraction explicitly before computing.
Pitfall 4: Applying the wrong transformation rule Under stress, swapping the rotation rule and the reflection rule produces completely wrong coordinates. For example, applying (x, y) → (−y, x) as a reflection instead of a 90° CCW rotation. Fix: Learn the transformation table by pattern: reflections flip one or both signs; rotations swap and change a sign. Review the table on paper before each test.
Related Entries
- Properties_Circles
- Similar_Triangles
- Lines_Angles_Parallel
- Right_Triangles_Pythagorean
- Linear_Equations
Quick Reference Card
| Formula | Expression |
|---|---|
| Distance | d = √[(x₂−x₁)² + (y₂−y₁)²] |
| Midpoint | M = ((x₁+x₂)/2, (y₁+y₂)/2) |
| Slope | m = (y₂−y₁)/(x₂−x₁) |
| Parallel | m₁ = m₂ |
| Perpendicular | m₁ × m₂ = −1 |
| Partition (ratio m:n from A) | P = A + m/(m+n) × (B−A) |
| Reflection / x-axis | (x, y) → (x, −y) |
| Reflection / y-axis | (x, y) → (−x, y) |
| Reflection / y=x | (x, y) → (y, x) |
| Rotation 90° CCW | (x, y) → (−y, x) |
| Dilation by k (origin) | (x, y) → (kx, ky) |
| None on reference sheet | Must memorize all of the above |