1.4 Coordinate geometry basics

Coordinate Geometry Fundamentals

Coordinate geometry connects algebra and geometry by letting you work with shapes using numbers and equations on a grid. Instead of just drawing and measuring, you can calculate exact distances, midpoints, and slopes to prove geometric properties.

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Cartesian coordinate system basics

The Cartesian coordinate system is a two-dimensional plane built from two perpendicular number lines:

• The x-axis runs horizontally (left-right position)
• The y-axis runs vertically (up-down position)

These axes cross at the origin, written as (0, 0). Every point on the plane is described by an ordered pair (x, y):

• The x-coordinate tells you horizontal position: negative values sit left of the origin, positive values sit right
• The y-coordinate tells you vertical position: negative values sit below the origin, positive values sit above

The axes divide the plane into four quadrants, labeled with Roman numerals and numbered counterclockwise starting from the upper right:

• Quadrant I (upper right): both coordinates positive (+, +)
• Quadrant II (upper left): x negative, y positive (−, +)
• Quadrant III (lower left): both coordinates negative (−, −)
• Quadrant IV (lower right): x positive, y negative (+, −)

Points that land directly on an axis don't belong to any quadrant. For example, (0, 5) is on the y-axis, not in Quadrant I or II.

Distance formula applications

The distance formula calculates the straight-line distance between any two points $(x_1, y_1)$ and $(x_2, y_2)$:

$d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$

This comes directly from the Pythagorean theorem. If you draw a right triangle using the two points, the horizontal leg has length $|x_2 - x_1|$, the vertical leg has length $|y_2 - y_1|$, and the hypotenuse is the distance between the points.

To find the distance between two points:

1. Identify the coordinates of both points
2. Substitute the x and y values into the formula
3. Square each difference, then add the results
4. Take the square root to get the distance

Example: Find the distance between (1, 2) and (4, 6).

$d = \sqrt{(4 - 1)^2 + (6 - 2)^2} = \sqrt{9 + 16} = \sqrt{25} = 5$

The distance formula is essential for calculating side lengths, perimeters, and diagonals of shapes plotted on the coordinate plane.

Midpoint formula calculations

The midpoint of a line segment is the point exactly halfway between its two endpoints. Given endpoints $(x_1, y_1)$ and $(x_2, y_2)$, the midpoint $(x_m, y_m)$ is found by averaging the coordinates:

$x_m = \frac{x_1 + x_2}{2} \qquad y_m = \frac{y_1 + y_2}{2}$

Think of it this way: to find the middle of two numbers, you add them and divide by 2. The midpoint formula just does that separately for x and y.

To find the midpoint of a segment:

1. Identify the coordinates of both endpoints
2. Add the two x-values and divide by 2 to get $x_m$
3. Add the two y-values and divide by 2 to get $y_m$
4. Write the result as the ordered pair $(x_m, y_m)$

Example: Find the midpoint of the segment from (−2, 3) to (6, 7).

$x_m = \frac{-2 + 6}{2} = 2 \qquad y_m = \frac{3 + 7}{2} = 5$

The midpoint is (2, 5). You can verify by checking that this point is equidistant from both endpoints using the distance formula.

Coordinate geometry for shapes

Once you can find distances, midpoints, and slopes, you can analyze and prove properties of geometric shapes on the coordinate plane.

Slope relationships are especially useful:

• Parallel lines have equal slopes. If two sides of a quadrilateral share the same slope, they're parallel.
• Perpendicular lines have slopes that are negative reciprocals of each other. Their slopes multiply to $-1$. For example, slopes of $\frac{2}{3}$ and $-\frac{3}{2}$ indicate perpendicularity.

Line equations let you describe lines algebraically:

• Slope-intercept form: $y = mx + b$, where $m$ is the slope and $b$ is the y-intercept. Best when you know or can read both values directly.
• Point-slope form: $y - y_1 = m(x - x_1)$, where $m$ is the slope and $(x_1, y_1)$ is any point on the line. More flexible when you're given a point and a slope.

Analyzing polygons on the coordinate plane typically involves combining these tools:

• Plot the vertices and use the distance formula to find side lengths (checking for congruent sides or computing perimeter)
• Use the midpoint formula to locate centers of sides, or to find where medians and diagonals intersect
• Calculate slopes of sides and diagonals to identify parallel sides (trapezoids, parallelograms) or right angles (rectangles, right triangles)

For example, to prove a quadrilateral is a rectangle, you'd show that opposite sides are equal in length (distance formula) and that consecutive sides are perpendicular (slopes are negative reciprocals).