Coordinate Geometry formulas

All 20 formulas in this topic

Slope Formulam=y2−y1x2−x1m = \frac{y_2 - y_1}{x_2 - x_1}The Slope Formula measures a line's steepness by dividing the vertical change between two points by their horizontal change.Slope-Intercept Formy=mx+by = mx + bSlope-intercept form directly reveals a line's slope and y-intercept, making it ideal for graphing and comparing linear relationships.Point-Slope Formy−y1=m(x−x1)y - y_1 = m(x - x_1)Point-Slope Form lets you write a line equation when you know the slope and one point; use it to quickly write equations or convert to standard form.Standard Form of a LineAx+By=CAx + By = CStandard form Ax + By = C expresses a line's equation using integer coefficients, making it ideal for finding intercepts and solving systems of equations.Distance Formulad=(x2−x1)2+(y2−y1)2d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}Calculates the shortest distance between two points on a coordinate plane using their coordinates and the Pythagorean theorem.Midpoint FormulaM=(x1+x22,;y1+y22)M = \left(\frac{x_1 + x_2}{2}, ; \frac{y_1 + y_2}{2}\right)The midpoint formula finds the coordinates of the point halfway between two endpoints on a line, given their x and y coordinates.Parallel and Perpendicular Slopesm∥=m,m⊥=−1mm_\parallel = m, \qquad m_\perp = -\frac{1}{m}Find whether two lines are parallel or perpendicular by comparing their slopes: identical for parallel, negative reciprocals for perpendicular.Equation of a Circle(x−h)2+(y−k)2=r2(x - h)^2 + (y - k)^2 = r^2Describes a circle using its center and radius; essential for finding circle equations, checking if points lie on circles, or solving circle geometry problems.Equation of a Parabola(x−h)2=4p(y−k)(x - h)^2 = 4p(y - k)The equation (x - h)^2 = 4p(y - k) describes a parabola with vertex at (h, k) and tells you which direction it opens and how wide it is.Equation of an Ellipse(x−h)2a2+(y−k)2b2=1\frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1Equation of an Ellipse describes an oval-shaped curve given its center, horizontal stretch, and vertical stretch; use it to model or graph any elliptical shape.Equation of a Hyperbola(x−h)2a2−(y−k)2b2=1\frac{(x-h)^2}{a^2} - \frac{(y-k)^2}{b^2} = 1The equation of a hyperbola describes all points forming a hyperbola with center at (h, k), used when modeling two-branch curves in coordinate geometry.x-Intercept and y-Intercepty=0  ⟹  x-intercept,x=0  ⟹  y-intercepty = 0 \implies x\text{-intercept}, \quad x = 0 \implies y\text{-intercept}Find where a line crosses the x-axis and y-axis by setting the other variable to zero—essential for graphing and understanding line behavior.Distance from a Point to a Lined=∣Ax0+By0+C∣A2+B2d = \frac{|Ax_0 + By_0 + C|}{\sqrt{A^2 + B^2}}Calculate the perpendicular distance from a point to a line given the line's equation in standard form and a point's coordinates.Section FormulaP=(mx2+nx1m+n,;my2+ny1m+n)P = \left(\frac{mx_2 + nx_1}{m+n}, ; \frac{my_2 + ny_1}{m+n}\right)Find a point that divides a line segment in a specific ratio, or use known division ratios to find missing endpoint coordinates.Area of a Triangle by CoordinatesA=12∣x1(y2−y3)+x2(y3−y1)+x3(y1−y2)∣A = \tfrac{1}{2}\left| x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2) \right|Find the area of any triangle when you know the coordinates of all three vertices, without needing to measure base and height.Graphing Linear Inequalitiesy>mx+b  ⟹  shade above a dashed liney > mx + b \implies \text{shade above a dashed line}Graphing linear inequalities shows which coordinate points satisfy a condition like y > mx + b by shading solution regions with appropriate line styles.Systems of Inequalities{y≤f(x)y≥g(x)\begin{cases} y \le f(x) \\ y \ge g(x) \end{cases}Find the region satisfying both inequalities by graphing on the same axes; use when a problem has multiple constraints that must all be satisfied.Line of Best Fity^=mx+b\hat{y} = mx + bA line that best approximates scattered data points, used to predict values and identify trends in relationships between two variables.Translating a Graphy=f(x−h)+ky = f(x - h) + kTranslate graphs using y = f(x - h) + k to shift any function horizontally by h and vertically by k units, needed when repositioning known curves.Reflecting a Graphy=−f(x) or y=f(−x)y = -f(x) \text{ or } y = f(-x)Reflecting a graph flips it across an axis—use it when you need to reverse a function's behavior, either top-to-bottom or left-to-right.

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