Analytic Geometry

<p>Pythagorean Theorem (B)</p> Signup and view all the answers

<p>$d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2}$ (A)</p> Signup and view all the answers

<p>Vertical change (B)</p> Signup and view all the answers

<p>Zero (B)</p> Signup and view all the answers

<p>They are equal. (A)</p> Signup and view all the answers

<p>If none of the other conditions are met (B)</p> Signup and view all the answers

<p>The coefficient of <em>x</em> in the line's equation (D)</p> Signup and view all the answers

<p>One is the negative reciprocal of the other. (C)</p> Signup and view all the answers

<p>To ensure the distance is positive (D)</p> Signup and view all the answers

<p>The ratios of the lengths of the segments (C)</p> Signup and view all the answers

<p>A slope of infinity (D)</p> Signup and view all the answers

<p>The x-coordinate of the midpoint of a line segment (D)</p> Signup and view all the answers

<p>$m = y_2 - y_1 / x_2 - x_1$ (A)</p> Signup and view all the answers

<p>Midpoint formula (A)</p> Signup and view all the answers

<p>-1/2 (A)</p> Signup and view all the answers

<p>The constant terms in the equations of the lines (B)</p> Signup and view all the answers

<p>Add the x-coordinates and add the y-coordinates (D)</p> Signup and view all the answers

A branch of Mathematics combining Algebra and Geometry to study shapes using coordinates.

The founder of Analytic Geometry who introduced the coordinate system in 1637.

A system where each point is represented by an ordered pair (x, y).

The formula to calculate the distance between two points (x1, y1) and (x2, y2).

Calculates distance between two points in 3D using (x, y, z) coordinates.

The perpendicular distance d of a point (x1, y1) to the line Ax + By + C = 0 is given by d = ±√(A^2 + B^2).

For finding distance, use + if B is positive and point is above/right; use - if B is negative and point is below/left.

The distance d between two parallel lines Ax + By + C1 = 0 and Ax + By + C2 = 0 is given by d = |C1 - C2| / √(A^2 + B^2).

To divide the segment between (x1, y1) and (x2, y2) at P with coordinates (x, y), use x = (x1r2 + x2r1) / (r1 + r2).

The y-coordinate of point P dividing the segment is y = (y1r2 + y2r1) / (r1 + r2).

The midpoint P of the segment joining (x1, y1) and (x2, y2) is x = (x1 + x2) / 2 and y = (y1 + y2) / 2.

The x-coordinate of the midpoint of a line segment is found using x = (x1 + x2) / 2.

The y-coordinate of the midpoint of a line segment is identified as y = (y1 + y2) / 2.

The ratio of vertical rise to horizontal run between two points on a line.

Lines with the same slope that never meet; slope of zero for horizontal lines, infinite for vertical lines.

Lines that intersect at right angles; the slopes are negative reciprocals of each other.

The slope (m) between points (x1, y1) and (x2, y2) is calculated as (y2 - y1)/(x2 - x1).

The angle θ formed by two lines with slopes m1 and m2 can be found using θ = tan⁻¹((m2 - m1) / (1 + m2 * m1)).

The formula to find the angle between two lines is θ = tan⁻¹((m2 - m1) / (1 + m2 * m1)).

The area of a triangle formed by points (x1, y1), (x2, y2), and (x3, y3) is calculated using determinants.

When determining area or angles, write vertices in counterclockwise order for proper calculation.