11.1 Distance and Midpoint Formulas; Circles

Distance and Midpoint Formulas; Circles

The distance and midpoint formulas give you two essential tools for working on the coordinate plane: measuring how far apart two points are, and finding the exact middle between them. Both formulas come straight from basic algebra and geometry you already know. In this section, you'll also use these ideas to write and graph equations of circles.

Distance and Midpoint Formulas

Distance Formula

The distance formula calculates the straight-line distance between two points $(x_1, y_1)$ and $(x_2, y_2)$ on the coordinate plane. It's derived directly from the Pythagorean Theorem: the horizontal difference between the points is one leg, the vertical difference is the other, and the distance is the hypotenuse.

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

Distance is always zero or positive. The order of the points doesn't matter, because squaring removes any negative signs.

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

1. Identify your two points: $(x_1, y_1) = (1, 2)$ and $(x_2, y_2) = (4, 6)$

2. Substitute into the formula: $d = \sqrt{(4 - 1)^2 + (6 - 2)^2}$

3. Simplify under the radical: $d = \sqrt{9 + 16} = \sqrt{25}$

4. Evaluate: $d = 5$

Midpoint Formula

The midpoint is the point that divides a line segment into two equal halves. To find it, you average the x-coordinates and average the y-coordinates of the endpoints.

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

Example: Find the midpoint of the segment connecting $(-3, 2)$ and $(5, 6)$.

1. Identify the endpoints: $(x_1, y_1) = (-3, 2)$ and $(x_2, y_2) = (5, 6)$
2. Average the x-coordinates: $\frac{-3 + 5}{2} = \frac{2}{2} = 1$
3. Average the y-coordinates: $\frac{2 + 6}{2} = \frac{8}{2} = 4$
4. The midpoint is $(1, 4)$

Think of it this way: the midpoint is just the "average point" of the two endpoints.

Circles

Standard Form of a Circle Equation

A circle is the set of all points that are a fixed distance (the radius) from a fixed point (the center). That definition translates directly into an equation using the distance formula.

The standard form of a circle with center $(h, k)$ and radius $r$ is:

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

• $(h, k)$ gives you the center. Watch the signs: in $(x - h)^2$, the value of $h$ is the opposite of what appears in the parentheses. So $(x + 2)^2$ means $h = -2$.
• The right side of the equation is $r^2$, not $r$. If the right side is 25, the radius is $\sqrt{25} = 5$.

Example: Write the equation of a circle with center $(-2, 3)$ and radius 5.

1. Substitute $h = -2$, $k = 3$, and $r = 5$ into the standard form.

2. $(x - (-2))^2 + (y - 3)^2 = 5^2$

3. Simplify: $(x + 2)^2 + (y - 3)^2 = 25$

Converting from general form: Sometimes you'll see a circle written in general form: $x^2 + y^2 + Cx + Dy + E = 0$. To convert this to standard form, group the $x$-terms and $y$-terms separately, then complete the square for each group. (You practiced completing the square back in your quadratics unit, and this is exactly why.)

Sketching Circles from Equations

Once you have the standard form, graphing is straightforward.

1. Identify the center $(h, k)$ and the radius $r = \sqrt{r^2}$ from the equation.

2. Plot the center on the coordinate plane.

3. Plot four guide points by moving $r$ units up, down, left, and right from the center:

   • $(h + r,\; k)$, $(h - r,\; k)$, $(h,\; k + r)$, $(h,\; k - r)$

4. Connect these four points with a smooth, round curve.

Example: Graph $(x)^2 + (y)^2 = 9$.

• Center: $(0, 0)$, Radius: $\sqrt{9} = 3$
• Four guide points: $(3, 0)$, $(-3, 0)$, $(0, 3)$, $(0, -3)$
• Plot and draw a smooth circle through those points.

A common mistake is forgetting to take the square root of the right side. If the equation says $= 9$, the radius is 3, not 9.