4. The Coordinate Plane

The coordinate plane is defined by two perpendicular number lines: the horizontal x-axis and the vertical y-axis, intersecting at the origin (0, 0). Every point is described by an ordered pair (x, y).

Distance formula: The distance between points (x1, y1) and (x2, y2) is:

• d = sqrt((x2-x1)^2 + (y2-y1)^2)

This comes directly from the Pythagorean theorem -- the horizontal and vertical differences are the legs of a right triangle.

Midpoint formula: The midpoint M of a segment from (x1, y1) to (x2, y2) is:

• M = ((x1+x2)/2, (y1+y2)/2)

Slope: The slope m of a line through two points is:

• m = (y2 - y1) / (x2 - x1) = rise / run
• Positive slope: line rises left to right. Negative slope: line falls left to right.
• Horizontal line: slope = 0. Vertical line: slope is undefined.
• Parallel lines have equal slopes. Perpendicular lines have slopes that are negative reciprocals: m1 * m2 = -1.

Equation of a circle: A circle with center (h, k) and radius r satisfies:

• (x - h)^2 + (y - k)^2 = r^2

To find the center and radius, rewrite the equation in this standard form by completing the square for both x and y.

Worked example: Find the center and radius of the circle x^2 + y^2 - 6x + 4y - 12 = 0.

• Group: (x^2 - 6x) + (y^2 + 4y) = 12.
• Complete the square: (x-3)^2 - 9 + (y+2)^2 - 4 = 12.
• Simplify: (x-3)^2 + (y+2)^2 = 25.
• Center: (3, -2); radius: sqrt(25) = 5.

• Slope-intercept form: y = mx + b (b is the y-intercept).
• Point-slope form: y - y1 = m(x - x1), useful when you know a point and slope.

⚠️ Common mistake: Forgetting that (x - h)^2 means h is positive if it appears as (x - 3). If the equation reads (x + 2)^2, the center's x-coordinate is -2, not +2.

💡 Tip: For distance problems on a grid, count the horizontal and vertical distances first, then apply a^2 + b^2 = c^2. This is faster than plugging four numbers into the distance formula.