What is the distance formula in coordinate geometry?

The distance formula is used to find the distance between two points (x₁, y₁) and (x₂, y₂) in the coordinate plane. It is given by: Distance = √[(x₂ - x₁)² + (y₂ - y₁)²].

How do you find the equation of a line given two points?

First, find the slope (m) using the formula: m = (y₂ - y₁) / (x₂ - x₁). Then, use the point-slope form of the equation of a line: y - y₁ = m(x - x₁), where (x₁, y₁) is one of the points.

What is the midpoint formula for two points?

The midpoint formula finds the midpoint M between two points (x₁, y₁) and (x₂, y₂). It is given by: M = ((x₁ + x₂)/2, (y₁ + y₂)/2).

Solve: Find the equation of a line passing through points (2, 3) and (4, 7). Hint: Find the slope first and then use point-slope form.

First, find the slope: m = (7 - 3) / (4 - 2) = 4 / 2 = 2. Now, use point-slope form with point (2, 3): y - 3 = 2(x - 2). Simplifying: y - 3 = 2x - 4, so y = 2x - 1.

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

The area A of a triangle with vertices at (x₁, y₁), (x₂, y₂), and (x₃, y₃) can be calculated using the formula: A = ½ |x₁(y₂ - y₃) + x₂(y₃ - y₁) + x₃(y₁ - y₂|.

How do you determine if three points are collinear?

Three points (x₁, y₁), (x₂, y₂), and (x₃, y₃) are collinear if the area of the triangle they form is zero, which can be checked using the area formula: A = ½ |x₁(y₂ - y₃) + x₂(y₃ - y₁) + x₃(y₁ - y₂| = 0.

Solve: Are the points (1, 2), (2, 3), and (3, 4) collinear? Hint: Use the area formula.

Substituting into the area formula: A = ½ |1(3 - 4) + 2(4 - 2) + 3(2 - 3)| = ½ |1(-1) + 2(2) + 3(-1)| = ½ |-1 + 4 - 3| = ½ |0| = 0. The points are collinear.

What is the slope-intercept form of a line?

The slope-intercept form of a line is given by: y = mx + b, where m is the slope and b is the y-intercept.

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

The slope (m) between two points (x₁, y₁) and (x₂, y₂) is calculated using the formula: m = (y₂ - y₁) / (x₂ - x₁).

Solve: Given points (3, 5) and (7, 9), find the slope of the line connecting these points. Hint: Use the slope formula.

Using the slope formula: m = (9 - 5) / (7 - 3) = 4 / 4 = 1. The slope of the line is 1.

What is the equation of a circle in standard form?

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

If the center of a circle is at (2, 3) and the radius is 4, what is the equation of the circle? Hint: Plug the values into the standard form.

Using the center (h, k) = (2, 3) and radius r = 4 in the standard form: (x - 2)² + (y - 3)² = 4². So, the equation is: (x - 2)² + (y - 3)² = 16.

What is the formula for the distance from a point to a line?

The distance d from a point (x₀, y₀) to the line Ax + By + C = 0 is given by: d = |Ax₀ + By₀ + C| / √(A² + B²).

Given the line 3x + 4y - 12 = 0 and point (2, 3), find the distance from the point to the line. Hint: Use the point-to-line distance formula.

Using the formula: d = |3(2) + 4(3) - 12| / √(3² + 4²) = |6 + 12 - 12| / √(9 + 16) = |6| / 5 = 6/5.