The distance formula helps you find how far apart two points are on a graph. It's like measuring with a ruler, but using math instead. The midpoint formula tells you where the exact middle of a line is. These tools are super useful for working with shapes and lines.

Circles are round shapes that follow special rules in math. You can describe a circle using an equation that tells you where its center is and how big it is. Drawing circles from these equations is like following a recipe to create a perfect round shape on your graph paper.

Distance and Midpoint Formulas

Distance formula for point length

• Calculates distance between two points $(x_1, y_1)$ and $(x_2, y_2)$ on coordinate plane
• Derived from Pythagorean Theorem $d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$
  • Always positive value or zero (distance from origin to itself)
• Applying formula:
  1. Identify coordinates of two points
  2. Substitute coordinates into formula
  3. Simplify expression under square root
  4. Calculate square root to find distance (3 units apart)

Midpoint formula for coordinates

• Point divides line segment into two equal parts
• Formula finds coordinates of midpoint $(x_m, y_m)$ given endpoints $(x_1, y_1)$ and $(x_2, y_2)$
  • x-coordinate of midpoint: $x_m = \frac{x_1 + x_2}{2}$
  • y-coordinate of midpoint: $y_m = \frac{y_1 + y_2}{2}$
• Calculating midpoint coordinates:
  1. Identify coordinates of endpoints ((-3, 2) and (5, 6))
  2. Substitute x-coordinates into x-coordinate formula and simplify
  3. Substitute y-coordinates into y-coordinate formula and simplify
  4. Resulting coordinates $(x_m, y_m)$ represent midpoint ((1, 4) is midpoint)

Circles

Standard form of circle equations

• Equation with center $(h, k)$ and radius $r$ is $(x - h)^2 + (y - k)^2 = r^2$
  • $(h, k)$ represents coordinates of center ((-2, 3) is center)
  • $r$ is length of radius (5 units long)
• Constructing equation:
  1. Identify center coordinates and radius length
  2. Substitute center coordinates for $h$ and $k$, radius length for $r$ in standard form equation
• Equation can be derived from general form $Ax^2 + By^2 + Cx + Dy + E = 0$
  • Convert from general to standard form by completing square for $x$ and $y$ terms
• The standard form is a quadratic equation in two variables

Sketching circles from equations

• Sketching on coordinate plane:
  1. Identify center coordinates $(h, k)$ and plot center point (origin (0, 0))
  2. Calculate points on circle using radius $r$
     • Add and subtract $r$ from both $h$ and $k$ to find four points: $(h + r, k)$, $(h - r, k)$, $(h, k + r)$, $(h, k - r)$ ((3, 0), (-3, 0), (0, 3), (0, -3))
  3. Plot four points and connect with smooth curve
• Analyzing from equations:
  • Center coordinates $(h, k)$ found by comparing equation to standard form
    • x-coordinate is opposite of x-term's coefficient divided by 2
    • y-coordinate is opposite of y-term's coefficient divided by 2
  • Radius $r$ found by taking square root of constant term on right side of equation

Coordinate System Basics

• The x-axis is the horizontal line in the coordinate plane
• The y-axis is the vertical line in the coordinate plane
• The origin is the point where the x-axis and y-axis intersect, with coordinates (0, 0)