What is the distance formula in coordinate geometry?

The distance formula calculates the distance between two points (x₁, y₁) and (x₂, y₂) in a Cartesian plane. It is given by the formula: d = √((x₂ - x₁)² + (y₂ - y₁)²).

What is the midpoint formula for a line segment between two points?

The midpoint of a line segment connecting two points (x₁, y₁) and (x₂, y₂) is given by the formula: M = ((x₁ + x₂) / 2, (y₁ + y₂) / 2).

How do you determine if three points (A, B, C) are collinear in a coordinate plane?

Three points A(x₁, y₁), B(x₂, y₂), and C(x₃, y₃) are collinear if the area of the triangle formed by them is zero. This can be checked using the determinant method: (x₁(y₂ - y₃) + x₂(y₃ - y₁) + x₃(y₁ - y₂) = 0).

What is the slope of a line passing through the points (2, 3) and (4, 7)?

The slope (m) of a line through two points (x₁, y₁) and (x₂, y₂) is calculated using the formula: m = (y₂ - y₁) / (x₂ - x₁). For the points (2, 3) and (4, 7), m = (7 - 3) / (4 - 2) = 4 / 2 = 2.

Find the equation of the line in slope-intercept form that passes through the point (1, 2) with a slope of 3.

What is the area of a triangle formed by the points (0, 0), (4, 0), and (0, 3)?

The area of a triangle can be calculated using the formula: Area = 1/2 * base * height. Here, base = 4 and height = 3, so Area = 1/2 * 4 * 3 = 6 square units.

What is the equation of a circle centered at (h, k) with radius r?

The equation of a circle with center (h, k) and radius r is given by (x - h)² + (y - k)² = r².

How can you find the y-intercept of the line 2x + 3y = 6?

To find the y-intercept, set x = 0 in the equation. This gives 2(0) + 3y = 6, leading to y = 2. Therefore, the y-intercept is (0, 2).

What is the section formula for finding a point that divides a line segment internally in the ratio m:n?

The coordinates of the point dividing the line segment joining (x₁, y₁) and (x₂, y₂) in the ratio m:n are given by: P = ((mx₂ + nx₁)/(m+n), (my₂ + ny₁)/(m+n)).

What is the formula for the slope of a line given two points?

The slope (m) of a line through two points (x₁, y₁) and (x₂, y₂) is given by: m = (y₂ - y₁) / (x₂ - x₁).

If the coordinates of point A are (a, b) and point B are (c, d), how do you find the distance between them?

The distance between points A(a, b) and B(c, d) is calculated using the distance formula: d = √((c - a)² + (d - b)²).

Calculate the area of a triangle with vertices (1, 2), (4, 6), and (6, 1).

The area can be calculated using the formula: Area = 1/2 | x₁(y₂ - y₃) + x₂(y₃ - y₁) + x₃(y₁ - y₂) |. Substituting the values gives Area = 1/2 | 1(6 - 1) + 4(1 - 2) + 6(2 - 6) | = 1/2 | 5 - 4 - 24 | = 1/2 * 25 = 12.5 square units.

A point P lies on the line 2x + 3y = 12. What are the coordinates of P if x = 3?

Substitute x = 3 into the equation: 2(3) + 3y = 12. This simplifies to 6 + 3y = 12, leading to 3y = 6, so y = 2. Therefore, the coordinates of point P are (3, 2).

What is the relationship between the slopes of two perpendicular lines?

If two lines are perpendicular, the product of their slopes (m₁ and m₂) is -1. Therefore, if m₁ * m₂ = -1, the lines are perpendicular.

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

The coordinates of the centroid (G) of a triangle are given by: G = ((x₁ + x₂ + x₃)/3, (y₁ + y₂ + y₃)/3). Substituting the values gives G = ((2 + 4 + 6)/3, (3 + 5 + 1)/3) = (4, 3).

How do you derive the area of a rectangle from its vertices?

The area of a rectangle can be calculated using the formula: Area = length * width. If the vertices of a rectangle are at (x₁, y₁), (x₂, y₂), (x₃, y₃), and (x₄, y₄), use the lengths of the sides parallel to the axes to determine the dimensions.

What are the x-intercept and y-intercept of the line 3x + 4y = 12?

To find the x-intercept, set y = 0: 3x = 12, giving x = 4. To find the y-intercept, set x = 0: 4y = 12, giving y = 3. Therefore, the intercepts are (4, 0) and (0, 3).

The line y = 2x + 3 intersects the x-axis at which point?

To find the x-intercept, set y = 0: 0 = 2x + 3. Solving gives x = -3/2. Therefore, the line intersects the x-axis at (-3/2, 0).

What is the area of the region defined by the inequalities x ≥ 0, y ≥ 0, and 2x + 3y ≤ 6?

First, find the intercepts of the line 2x + 3y = 6, which are (3, 0) and (0, 2). The area of the triangle formed in the first quadrant is given by: Area = 1/2 * base * height = 1/2 * 3 * 2 = 3 square units.