Coordinate Geometry

The distance formula calculates the distance between two points (x1, y1) and (x2, y2) in a plane. It is derived from the Pythagorean theorem, resulting in √[(x2 - x1)² + (y2 - y1)²]. For example, the distance between (1, 2) and (4, 6) is √[(4-1)² + (6-2)²] = 5 units.

The midpoint is found by averaging the x-coordinates and y-coordinates of the two points. The formula is [(x1 + x2)/2, (y1 + y2)/2]. For instance, the midpoint between (2, 3) and (4, 7) is [(2+4)/2, (3+7)/2] = (3, 5).

The section formula finds the coordinates of a point dividing a line segment internally in a given ratio m:n. The formula is [(mx2 + nx1)/(m+n), (my2 + ny1)/(m+n)]. For example, dividing the segment joining (1, 2) and (4, 5) in the ratio 2:1 gives [(2*4 + 1*1)/3, (2*5 + 1*2)/3] = (3, 4).

Three points are collinear if the area formed by them is zero. Calculate the area using 1/2 |x1(y2 - y3) + x2(y3 - y1) + x3(y1 - y2)|. If the result is zero, the points are collinear. For example, (1, 1), (2, 2), and (3, 3) are collinear.

The slope indicates the steepness and direction of a line. It is calculated as (y2 - y1)/(x2 - x1). A positive slope means the line rises, while a negative slope means it falls. For example, the slope between (1, 1) and (3, 5) is (5-1)/(3-1) = 2.

The area is calculated using 1/2 |x1(y2 - y3) + x2(y3 - y1) + x3(y1 - y2)|. For example, for vertices (0, 0), (4, 0), and (0, 3), the area is 1/2 |0(0-3) + 4(3-0) + 0(0-0)| = 6 square units.

Two lines are parallel if their slopes are equal. For example, lines with slopes 2 and 2 are parallel, while lines with slopes 2 and -2 are not.

Two lines are perpendicular if the product of their slopes is -1. For example, lines with slopes 2 and -0.5 are perpendicular because 2 * -0.5 = -1.

The centroid is the average of the vertices' coordinates. The formula is [(x1 + x2 + x3)/3, (y1 + y2 + y3)/3]. For a triangle with vertices (1, 2), (3, 4), and (5, 6), the centroid is at (3, 4).

Coordinate geometry is used in navigation, architecture, and physics to model and solve real-world problems. For example, GPS systems use it to determine locations, and architects use it to design structures.

Check if all sides are equal and diagonals are equal and bisect each other at 90 degrees. For example, for points (0, 0), (2, 0), (2, 2), and (0, 2), sides are 2 units each, and diagonals are 2√2 units, confirming it's a square.

The distance of a point (x, y) from the origin (0, 0) is √(x² + y²). For example, the distance of (3, 4) from the origin is √(3² + 4²) = 5 units.

Use the section formula in reverse. If point P divides AB, the ratio is (Px - Ax)/(Bx - Px) and (Py - Ay)/(By - Py). For example, if P(2, 3) divides A(1, 2) and B(4, 5), the ratio is (2-1)/(4-2) = 1:2.

Internal division means the point lies between the two endpoints, while external division means it lies outside. The section formula adjusts signs accordingly. For example, dividing (1, 2) and (4, 5) externally in 2:1 gives [(2*4 - 1*1)/(2-1), (2*5 - 1*2)/(2-1)] = (7, 8).

Set the distances from the point to each given point equal and solve. For example, a point (x, y) equidistant from (1, 2) and (3, 4) satisfies √[(x-1)² + (y-2)²] = √[(x-3)² + (y-4)²], simplifying to x + y = 5.

A positive slope indicates the line rises to the right, while a negative slope falls to the right. A zero slope is horizontal, and an undefined slope is vertical. For example, y = 2x + 1 has a positive slope of 2.

Check if the midpoints of the diagonals are the same. For points A(1, 2), B(4, 3), C(5, 6), D(2, 5), the midpoint of AC is (3, 4) and BD is (3, 4), confirming a parallelogram.

Divide it into two triangles and sum their areas. For vertices (x1, y1), (x2, y2), (x3, y3), (x4, y4), the area is 1/2 |x1y2 + x2y3 + x3y4 + x4y1 - (y1x2 + y2x3 + y3x4 + y4x1)|.

First, find the slope (m) as (y2 - y1)/(x2 - x1). Then, use point-slope form y - y1 = m(x - x1). For points (1, 2) and (3, 4), the slope is 1, so the equation is y - 2 = 1(x - 1) or y = x + 1.

A point lies on the x-axis if its y-coordinate is zero, and on the y-axis if its x-coordinate is zero. For example, (3, 0) is on the x-axis, and (0, 4) is on the y-axis.

Reflect over the x-axis by changing the sign of the y-coordinate, and over the y-axis by changing the sign of the x-coordinate. For example, reflecting (2, 3) over the x-axis gives (2, -3), and over the y-axis gives (-2, 3).

The distance formula is fundamental for calculating lengths, verifying geometric shapes, and solving real-world problems like finding the shortest path. It's derived from the Pythagorean theorem and applies universally in plane geometry.

Calculate the lengths of all sides using the distance formula. If all sides are equal, it's equilateral; if two sides are equal, it's isosceles; if all sides are different, it's scalene. Also, check if it satisfies the Pythagorean theorem for a right triangle.

The centroid is the intersection point of the medians, calculated as [(x1 + x2 + x3)/3, (y1 + y2 + y3)/3]. It's the triangle's balance point and is used in physics and engineering for center of mass calculations.