What is the distance formula between two points?

The distance between two points (x₁, y₁) and (x₂, y₂) is given by the formula: distance = √((x₂ - x₁)² + (y₂ - y₁)²). For example, to find the distance between (2, 3) and (5, 7), calculate: distance = √((5 - 2)² + (7 - 3)²) = √(9 + 16) = √25 = 5.

What is the slope formula of a line passing through two points?

The slope (m) of a line passing through the points (x₁, y₁) and (x₂, y₂) is calculated as: m = (y₂ - y₁) / (x₂ - x₁). For example, if the points are (1, 2) and (4, 6), the slope is m = (6 - 2) / (4 - 1) = 4 / 3.

What is a quick trick to determine if three points are collinear?

To check if three points (x₁, y₁), (x₂, y₂), and (x₃, y₃) are collinear, calculate the slopes of the pairs: slope between (x₁, y₁) and (x₂, y₂) and between (x₂, y₂) and (x₃, y₃). If the slopes are equal, then the points are collinear.

Find the equation of a line that passes through the points (2, 3) and (5, 7).

What is the Midpoint Formula?

The midpoint M of a line segment connecting points (x₁, y₁) and (x₂, y₂) is given by M = ((x₁ + x₂) / 2, (y₁ + y₂) / 2). For example, the midpoint between (2, 3) and (4, 7) is M = ((2 + 4) / 2, (3 + 7) / 2) = (3, 5).

What is the formula for the area of a triangle given its vertices?

What is the equation of a circle with center at the origin and radius r?

The equation of a circle centered at the origin (0, 0) with radius r is x² + y² = r². For example, if the radius is 6, the equation is x² + y² = 36.

Find the coordinates of the centroid of triangle with vertices at (1, 2), (3, 4), and (5, 6).

The coordinates of the centroid G of a triangle with vertices (x₁, y₁), (x₂, y₂), and (x₃, y₃) is G = ((x₁ + x₂ + x₃)/3, (y₁ + y₂ + y₃)/3). Thus, G = ((1 + 3 + 5)/3, (2 + 4 + 6)/3) = (3, 4).

If two lines are perpendicular, what can be said about their slopes?

If two lines are perpendicular, the product of their slopes (m₁ and m₂) is -1: m₁ * m₂ = -1. For example, if one line has a slope of 3, the slope of the perpendicular line will be -1/3.

What is the area of the triangle with vertices at (0,4), (3,6), and (-8,-2)?

Using the area formula A = 1/2 | x₁(y₂ - y₃) + x₂(y₃ - y₁) + x₃(y₁ - y₂) |, we substitute to find A = 1/2 | 0(6 - (-2)) + 3((-2) - 4) + (-8)(4 - 6) | = 1 square unit.