Analytic Geometry: Coordinate Systems and Equations

What fundamental principle underlies calculating the distance between two points, both in a plane and in space?

In analytic geometry, how are points represented within the Cartesian coordinate system?

Suppose you have two points in space, A(1, 2, 3) and B(4, 5, 6). What is the distance between points A and B?

Which of the following statements accurately describes Rene Descartes' contribution to the field of mathematics?

In a two-dimensional Cartesian coordinate system, what term is used to describe the horizontal coordinate of a point?

Line A passes through points (1, 5) and (3, 9). Line B is perpendicular to Line A. What is the slope of Line B?

A line is parallel to the x-axis. What is its slope?

Line P has a slope of 3. Line Q has a slope of -1/3. What is the relationship between Line P and Line Q?

Given two lines with slopes $m_1 = 2$ and $m_2 = 5$, what is the tangent of the angle between these two lines?

Which condition indicates that two lines are parallel?

What is the slope of a line parallel to the y-axis?

Three points A(1, 2), B(4, 7), and C(6, 3) form a triangle. If the coordinates are listed in a counterclockwise direction, which order is correct for using the area formula?

A triangle has vertices at (0, 0), (1, 1), and (2, 0). If calculating the area using coordinate geometry, what is the correct sequential order of these points?

A point (1, 2) is located near a line defined by $3x + 4y + 5 = 0$. According to the convention provided, which sign should be used in the point-to-line distance formula?

Two parallel lines are defined by the equations $2x + 3y + 6 = 0$ and $2x + 3y + 12 = 0$. What is the correct setup to calculate the distance d between them?

A line segment is defined by the points (1, 5) and (4, 2). Point P divides this segment in a 2:1 ratio. Which formula accurately calculates the x-coordinate of point P?

A line segment has endpoints at (8, -2) and (4, 6). What are the coordinates of the midpoint of this segment?

Consider a line segment with endpoints A(2, -3) and B(5, 1). If point P divides AB in a 1:2 ratio, find the y-coordinate of point P.

Given two parallel lines defined by $4x - 3y + 8 = 0$ and $4x - 3y - 7 = 0$, calculate the distance between them.

A point (5, -1) is near the line $-2x + 5y - 3 = 0$. Determine the correct sign to use in the point-to-line distance formula according to the provided conventions.

Find the midpoint of a line segment with endpoints at (-3, 7) and (5, -9).