Coordinate Geometry and Algebra Review

Vocabulary flashcards covering formulas, definitions, and procedures for coordinate geometry, line equations, and circle equations from the lecture notes.

Point-Slope Form

The linear equation formula given as $y - y_1 = m(x - x_1)$, used to determine the equation of a line passing through given points before rearranging into standard form.

Slope-Intercept Formula ($y = mx + c$)

A straight line formula where $m$ represents the slope and $c$ represents the y-intercept.

Radius Formula (Distance Formula)

The formula described in the notes as the square root of $(X_2 - X_1) + (Y_2 - Y_1)$, used to find side lengths of triangles or the radius of a circle from coordinate points.

Right-Angled Triangle Verification

The method of calculating the lengths of sides $AB$, $BC$, and $AC$ using the distance formula; if two sides equal one side, the triangle is confirmed to be right-angled.

Circle Center from Diameter Endpoints

The coordinate determined by adding the two x-coordinates of the diameter endpoints and dividing by $2$, and performing the same operation on the y-coordinates.

Circle Equation Form

The formula structure given in the notes as $(X - h)^2 + (Y - K)^2$, where the center coordinates and radius are substituted.

Completing the Square for Circle Equations

The process of grouping x-terms and y-terms separately, dividing the linear coefficient in each group by $2$, squaring that number, and adding it to both the respective group and the constant on the other side of the equation.

Center from General Circle Form ($x^2 + y^2 - 8x + 6y - 3 = 0$)

The point found by taking the numbers inside the factored squared x and y groups and switching their signs to form the $(x, y)$ coordinates.

Radius from General Circle Form ($x^2 + y^2 - 8x + 6y - 3 = 0$)

The value obtained by taking the final accumulated number on the right-hand side of the completed-square equation and placing a square root on it.

K-Substitution Method for Parallel/Perpendicular Lines

The technique of rewriting an equation like $5x + 2y - 3 = 0$ with a positive $K$ in place of the constant, substituting given coordinates $(1, -2)$ to solve for $K$, and re-inserting $K$ into the final standard equation.

Quadratic Formula Step for Slope Equations

The solving procedure applied after equating the slope formula to a rational expression like $\frac{2x+1}{x+7}$, cross-multiplying, and moving all terms to the left side with switched signs, which yields two solutions.