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 a Cartesian plane. It is given by the formula: d = √((x2 - x1)² + (y2 - y1)²).

What is the midpoint formula for two points?

The midpoint formula calculates the midpoint M between two points (x1, y1) and (x2, y2) as M = ((x1 + x2)/2, (y1 + y2)/2).

How do you determine the slope of a line given two points?

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

Find the equation of a line with a slope of 2 that passes through the point (3, 4).

Hint: Use the point-slope form of a line.

Using the point-slope form, y - y1 = m(x - x1), where m = 2, x1 = 3, and y1 = 4: y - 4 = 2(x - 3). Simplifying gives the equation: y = 2x - 2.

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

The slopes of two perpendicular lines are negative reciprocals of each other. If m1 is the slope of the first line, and m2 is the slope of the second line, then m1 * m2 = -1.

What is the equation of a circle with center (h, k) and 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 do you find the equation of a line in slope-intercept form?

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 the area of a triangle formed by three points in the coordinate plane?

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

What is the standard form of a linear equation?

The standard form of a linear equation is Ax + By = C, where A, B, and C are integers, and A ≥ 0.

Determine the coordinates of the intersection of the lines y = 3x + 2 and y = -x + 4.

Hint: Set the two equations equal to each other.

To find the intersection, set 3x + 2 = -x + 4. Solving gives x = ½. Substituting x back into either equation gives y = 3(½) + 2 = 3. The intersection point is (½, 3).

What is the slope of a horizontal line?

The slope of a horizontal line is 0, as there is no change in the y-value regardless of the change in the x-value.

What is the slope of a vertical line?

The slope of a vertical line is undefined, as the x-value does not change while the y-value can vary.

If a line has a slope of 4 and passes through the origin, what is its equation?

The equation of the line is y = 4x, since it passes through the origin (0, 0), giving a y-intercept of 0.

What is the formula to find the distance from a point (x0, y0) to a line Ax + By + C = 0?

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

Find the length of the line segment between the points (2, 3) and (5, 7).

Hint: Use the distance formula.

Using the distance formula d = √((x2 - x1)² + (y2 - y1)²), we have d = √((5 - 2)² + (7 - 3)²) = √(3² + 4²) = √(9 + 16) = √25 = 5.