Coordinate Geometry: Points, Distance, Midpoint, and Circles

Coordinate Geometry

Plotting Points on the Cartesian Coordinate System

The Cartesian Coordinate System allows us to represent points in a plane using ordered pairs. Each point is defined by its x-coordinate (horizontal axis) and y-coordinate (vertical axis).

• Ordered Pair: Written as (x, y), where x is the horizontal value and y is the vertical value.

• Example: The point (3,2) is plotted by moving 3 units along the x-axis and 2 units along the y-axis.

Note: When you plot a point and draw lines to the axes, you form a rectangle whose sides are determined by the coordinates.

Midpoint Formula

To find the point exactly halfway between two given points, use the midpoint formula:

• Formula:

$\left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)$

• Example: Find the midpoint between (1,2) and (5,8):

$\left( \frac{1+5}{2}, \frac{2+8}{2} \right) = (3,5)$

Distance Formula

The distance formula calculates the straight-line distance between two points in the plane:

• Formula:

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

• Example: Find the distance between (1,5) and (5,8):

$d = \sqrt{(5-1)^2 + (8-5)^2} = \sqrt{16 + 9} = \sqrt{25} = 5$

Geometric Interpretation: The distance is the length of the hypotenuse of a right triangle formed by the horizontal and vertical differences between the points. This is an application of the Pythagorean Theorem: $a^2 + b^2 = c^2$.

Equations of Circles

Definition and Standard Form

A circle is the set of all points in a plane that are a fixed distance (the radius) from a fixed point (the center).

• Standard Form of the Equation of a Circle:

$(x - h)^2 + (y - k)^2 = r^2$

• Where (h, k) is the center and r is the radius.

• Example: For $(x-2)^2 + (y+3)^2 = 25$, the center is (2, -3) and the radius is 5.

Hint: Be careful with signs for the center; a double negative is positive.

Finding the Center and Radius from General Form

Sometimes, the equation of a circle is not in standard form. To convert it, use completing the square:

• Example: Find the center and radius of $x^2 + 4x + y^2 = 8$.

1. Group x and y terms: $x^2 + 4x + y^2 = 8$

2. Complete the square for x: $x^2 + 4x + 4 + y^2 = 8 + 4$

3. Rewrite: $(x+2)^2 + y^2 = 12$

4. Center: (-2, 0), Radius: $\sqrt{12} = 2\sqrt{3}$

• Another Example: $x^2 - 6x + y^2 + 8y = 3$

• Complete the square for both variables:

• $(x^2 - 6x + 9) + (y^2 + 8y + 16) = 3 + 9 + 16$

• $(x-3)^2 + (y+4)^2 = 28$

• Center: (3, -4), Radius: $\sqrt{28} = 2\sqrt{7}$

Summary Table: Key Formulas in Coordinate Geometry

Concept	Formula	Description
Midpoint	$\left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)$	Point halfway between two points
Distance	$d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$	Straight-line distance between two points
Circle (Standard Form)	$(x - h)^2 + (y - k)^2 = r^2$	Equation of a circle with center (h, k) and radius r