Coordinate Geometry

The Coordinate Plane

The coordinate plane consists of two perpendicular number lines: the horizontal $x$-axis and the vertical $y$-axis. They intersect at the origin $(0, 0)$. Every point is described by an ordered pair $(x, y)$, where $x$ is the horizontal distance from the origin and $y$ is the vertical distance.

The four quadrants are:

• Quadrant I: $x > 0$, $y > 0$ (upper right)
• Quadrant II: $x < 0$, $y > 0$ (upper left)
• Quadrant III: $x < 0$, $y < 0$ (lower left)
• Quadrant IV: $x > 0$, $y < 0$ (lower right)

The ACT may ask which quadrant a point lies in, or which quadrants a line passes through. Knowing the sign conventions is essential.

Slope

The slope of a line measures its steepness and direction. Given two points $(x_1, y_1)$ and $(x_2, y_2)$:

$m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{\text{rise}}{\text{run}}$

Key slope facts every ACT student must know:

• Positive slope: line rises from left to right (goes uphill)
• Negative slope: line falls from left to right (goes downhill)
• Zero slope: horizontal line ($y = c$) — the rise is zero
• Undefined slope: vertical line ($x = c$) — the run is zero (division by zero)

The slope tells you how much $y$ changes for every 1-unit change in $x$. A slope of $\frac{3}{4}$ means "up 3, right 4."

Equations of Lines

Slope-intercept form: $y = mx + b$, where $m$ is the slope and $b$ is the $y$-intercept (the point where the line crosses the $y$-axis). This is the most useful form for the ACT because it immediately reveals both the slope and the intercept.

Point-slope form: $y - y_1 = m(x - x_1)$ — useful when you know a specific point $(x_1, y_1)$ and the slope $m$. You can always convert this to slope-intercept form by distributing and simplifying.

Standard form: $Ax + By = C$, where $A$, $B$, and $C$ are integers. The ACT often presents equations this way. To find the slope from standard form, rearrange to slope-intercept form or use the shortcut: $m = -\frac{A}{B}$.

Finding the equation of a line: The most common ACT question type asks you to find the equation of a line given two points, or a point and a slope.

1. Find the slope using $m = \frac{y_2 - y_1}{x_2 - x_1}$
2. Plug the slope and one point into point-slope form
3. Simplify to slope-intercept form if needed

The Distance Formula

The distance between two points $(x_1, y_1)$ and $(x_2, y_2)$ is:

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

This formula is simply the Pythagorean theorem applied to the coordinate plane. The horizontal distance $|x_2 - x_1|$ and vertical distance $|y_2 - y_1|$ form the legs of a right triangle, and the straight-line distance is the hypotenuse. If you ever forget the distance formula, draw the right triangle and use $a^2 + b^2 = c^2$.

Pro tip: Before computing the full square root, check if the numbers form a Pythagorean triple. If the legs are 6 and 8, the hypotenuse is 10 (the $3$-$4$-$5$ triple scaled by 2). Recognizing triples saves significant time.

The Midpoint Formula

The midpoint of the segment connecting $(x_1, y_1)$ and $(x_2, y_2)$ is:

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

Think of it as averaging the $x$-coordinates and averaging the $y$-coordinates. The midpoint is exactly halfway between the two points in both directions.

A useful extension: if you know one endpoint and the midpoint, you can find the other endpoint. If $(x_1, y_1)$ and midpoint $(m_x, m_y)$ are known, then $x_2 = 2m_x - x_1$ and $y_2 = 2m_y - y_1$.

Parallel and Perpendicular Lines

These relationships are among the most frequently tested concepts in ACT coordinate geometry:

• Parallel lines have the same slope: $m_1 = m_2$. They never intersect.
• Perpendicular lines have slopes that are negative reciprocals: $m_1 \cdot m_2 = -1$, or equivalently $m_2 = -\frac{1}{m_1}$.

For example, if a line has slope $\frac{2}{3}$:

• A parallel line also has slope $\frac{2}{3}$
• A perpendicular line has slope $-\frac{3}{2}$ (flip the fraction and negate)

Special cases: A horizontal line ($m = 0$) is perpendicular to a vertical line (undefined slope).

Intercepts

The $x$-intercept is the point where the line crosses the $x$-axis. At this point, $y = 0$. To find it, set $y = 0$ in the equation and solve for $x$.

The $y$-intercept is the point where the line crosses the $y$-axis. At this point, $x = 0$. To find it, set $x = 0$ (or read the value of $b$ directly from $y = mx + b$).

Graphing Linear Equations

To graph $y = mx + b$:

1. Plot the $y$-intercept $(0, b)$
2. Use the slope $m = \frac{\text{rise}}{\text{run}}$ to find a second point (from the $y$-intercept, go up "rise" units and right "run" units)
3. Draw the line through both points

For horizontal lines $y = c$: draw a horizontal line at height $c$.

For vertical lines $x = c$: draw a vertical line at position $c$.

Circles in the Coordinate Plane

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

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

Watch the signs: In $(x - 3)^2 + (y + 1)^2 = 25$, the center is $(3, -1)$ and the radius is $\sqrt{25} = 5$. The $+1$ inside the parentheses with $y$ means $k = -1$ (the sign flips).

To determine whether a point lies inside, on, or outside a circle, compute the distance from the point to the center and compare it to the radius:

• If distance $< r$: inside
• If distance $= r$: on the circle
• If distance $> r$: outside

Tip 1: Sketch It Out

Many coordinate geometry problems become dramatically easier with a quick sketch. Even a rough diagram on your test booklet helps you visualize relationships, estimate answers, and eliminate wrong choices. Spend 10 seconds sketching — it can save you minutes of confusion.

Tip 2: Memorize the Formulas

The ACT does not provide a formula sheet. You must have slope, distance, midpoint, and the circle equation memorized cold before test day. Consider writing them on your scratch paper in the first 30 seconds of the section.

Tip 3: Use the Answer Choices to Estimate

If the question asks for the distance between two points and the choices are 5, 10, 13, 15, and 17, you can sometimes estimate by counting grid squares on your sketch rather than doing the full calculation. This is especially helpful when you are short on time.

Tip 4: Convert to Slope-Intercept Form Immediately

When an equation is given in standard form ($Ax + By = C$), your first move should be converting to $y = mx + b$. This takes a few seconds and makes slope, intercept, and comparison problems trivial.

Tip 5: Remember the Special Relationships

Parallel = same slope. Perpendicular = negative reciprocal. These two facts alone can answer many ACT questions in under 30 seconds. Drill them until they are automatic.