Distance and Midpoint Formulas; Circles

Distance and Midpoint Formulas; Circles

Distance Formula

The distance formula is used to find the distance between two points in the coordinate plane. It is derived from the Pythagorean Theorem.

• Formula:

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

• Where: $(x_1, y_1)$ and $(x_2, y_2)$ are the coordinates of the two points.

• Example: Find the distance between $(-1, -3)$ and $(2, 3)$.

$d = \sqrt{(2 - (-1))^2 + (3 - (-3))^2} = \sqrt{(3)^2 + (6)^2} = \sqrt{9 + 36} = \sqrt{45} = 3\sqrt{5} \approx 6.71$

Midpoint Formula

The midpoint formula finds the point exactly halfway between two given points in the coordinate plane.

• Formula:

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

• Where: $(x_1, y_1)$ and $(x_2, y_2)$ are the endpoints of the segment.

• Example: Find the midpoint of the segment with endpoints $(1, 2)$ and $(7, -3)$.

$M = \left( \frac{1 + 7}{2}, \frac{2 + (-3)}{2} \right) = \left( \frac{8}{2}, \frac{-1}{2} \right) = (4, -0.5)$

Circles

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

• Formula:

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

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

• Example: Write the standard form of the equation of the circle with center $(0, -6)$ and radius $10$.

$(x - 0)^2 + (y + 6)^2 = 10^2$ $x^2 + (y + 6)^2 = 100$

• Example: Find the center and radius of the circle whose equation is $x^2 + (y + 6)^2 = 100$.

Center: $(0, -6)$ Radius: $\sqrt{100} = 10$

General Form of the Equation of a Circle

• The general form of a circle's equation is:

$x^2 + y^2 + Dx + Ey + F = 0$

• To convert from general form to standard form, complete the square for both $x$ and $y$ terms.

• Example: Write in standard form: $(x + 3)^2 + (y - 1)^2 = 4$

This equation is already in standard form with center $(-3, 1)$ and radius $2$.

• Example: Convert $x^2 + y^2 + 4x - 4y - 1 = 0$ to standard form.

Step 1: Group $x$ and $y$ terms: $(x^2 + 4x) + (y^2 - 4y) = 1$ Step 2: Complete the square:

$x^2 + 4x + 4 + y^2 - 4y + 4 = 1 + 4 + 4$ $(x + 2)^2 + (y - 2)^2 = 9$

Center: $(-2, 2)$ Radius: $3$