What is the difference between the distance and midpoint formulas?

The distance formula, d = sqrt((x2 - x1)^2 + (y2 - y1)^2), gives the length of a segment. The midpoint formula, ((x1 + x2)/2, (y1 + y2)/2), gives the point exactly halfway between the endpoints. One measures length; the other locates a point.

How do you find the slope of a perpendicular line on the ACT?

Take the negative reciprocal of the original slope: flip the fraction and change its sign so the two slopes multiply to -1. For example, a line with slope 2 has perpendicular slope -1/2, and a line with slope -3/4 has perpendicular slope 4/3.

Why is the slope of a vertical line undefined?

A vertical line has no horizontal change, so the run (x2 - x1) is 0. Slope is rise over run, and dividing by zero is undefined. A horizontal line is the opposite case: its rise is 0, giving a slope of exactly 0.

How can coordinate geometry show that a triangle is a right triangle?

Compute the slopes of two sides that meet at a vertex. If those slopes are negative reciprocals (their product is -1), the sides are perpendicular and that angle is a right angle, making the triangle a right triangle.

CramClub • Coordinate Geometry — ACT® Guide & Practice

$m = \dfrac{y_2 - y_1}{x_2 - x_1}$

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

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

$m_1 \cdot m_2 = -1$

The xy-Plane Is a Measuring Tool

Coordinate geometry puts algebra and geometry in the same room. Every point is an ordered pair $(x, y)$ , and the four workhorse ideas — slope, distance, midpoint, and how lines meet at parallel or perpendicular angles — let you measure a figure instead of just looking at it. On the ACT you will see these both as bare number problems and folded into figures, so knowing which formula answers which question is half the battle.

The key mental move: a segment between two points is really the hypotenuse of a right triangle whose legs are the horizontal change (the run, $\Delta x$ ) and the vertical change (the rise, $\Delta y$ ). Slope compares those legs, distance combines them, and the midpoint averages the endpoints.

Slope — Steepness and Direction

The slope $m$ measures how fast $y$ changes as $x$ increases:

$m = \frac{\Delta y}{\Delta x} = \frac{y_2 - y_1}{x_2 - x_1}$

A positive slope rises left to right; a negative slope falls. A horizontal line has slope $0$ (no rise), and a vertical line has an undefined slope (the run is $0$ , and you cannot divide by zero). Keep the order of the points consistent: if $y_2$ comes from the second point, $x_2$ must too. Reversing one but not the other flips the sign.

Distance — The Pythagorean Theorem in Disguise

The distance between two points is the length of that hypotenuse:

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

Because both differences are squared, the order of subtraction does not matter — $(x_2 - x_1)^2 = (x_1 - x_2)^2$ . If either coordinate is negative, subtracting it adds: the run from $x = -3$ to $x = 4$ is $4 - (-3) = 7$ . That double-negative is the single most common place students lose a point.

Midpoint — Just an Average

The midpoint of a segment is the average of the endpoints in each direction:

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

A useful variation the ACT loves: if you are told the midpoint and one endpoint, work backwards. The midpoint is halfway, so the other endpoint is the same jump again — double the midpoint coordinate and subtract the known endpoint: $x_2 = 2x_M - x_1$ .

Parallel and Perpendicular Lines

Slope also tells you how two lines relate.

Relationship	Slope condition
Parallel	Equal slopes: $m_1 = m_2$
Perpendicular	Negative reciprocals: $m_1 \cdot m_2 = -1$

To find a perpendicular slope, flip the fraction and change the sign. The negative reciprocal of $\tfrac{2}{3}$ is $-\tfrac{3}{2}$ ; the negative reciprocal of $-4$ (that is, $-\tfrac{4}{1}$ ) is $\tfrac{1}{4}$ . A horizontal line ( $m = 0$ ) is perpendicular to a vertical line (undefined slope) — that pair is the one exception the reciprocal rule cannot state numerically.

Putting It on a Graph

When a figure is drawn, read coordinates straight off the grid, then apply the right tool:

How steep / which way? → slope.
How long is a segment? → distance.
Where is the center of a segment? → midpoint.
Do two sides make a right angle? → check that their slopes multiply to $-1$ .

That last test is how the ACT asks you to prove a triangle is right or a quadrilateral is a rectangle without any protractor.

A Reliable Game Plan

Label the points $(x_1, y_1)$ and $(x_2, y_2)$ before plugging in, and keep the order fixed.
Watch every negative coordinate: subtracting a negative adds.
For "other endpoint" problems, use $2x_M - x_1$ rather than re-deriving the average.
To classify lines, compare slopes only — the intercepts are irrelevant.

Common Mistakes to Avoid

Inverting slope as $\tfrac{\Delta x}{\Delta y}$ (run over rise) instead of rise over run.
Forgetting to square inside the distance formula, or taking the root of each term separately.
Using the plain reciprocal — not the negative reciprocal — for a perpendicular slope.
Averaging only the $x$ -coordinates and forgetting the $y$ for a midpoint.

What is the slope of the line through $(1, 2)$ and $(5, 14)$ ?

Find the midpoint of the segment joining $(-3, 4)$ and $(7, -2)$ .

Find the distance between $(-2, 1)$ and $(6, 7)$ .

The midpoint of segment $\overline{AB}$ is $(4, -1)$ . If $A = (1, 3)$ , find the coordinates of $B$ .

A line passes through $(0, 5)$ and $(4, -1)$ . What is the slope of any line perpendicular to it?

Triangle $PQR$ has vertices $P(1, 1)$ , $Q(4, 7)$ , and $R(7, 1)$ . Is angle $P$ a right angle?

Writing slope as run over rise, $\dfrac{x_2 - x_1}{y_2 - y_1}$ , which inverts the answer.

Slope is rise over run: the $y$ -difference is on top. A quick check — a steep line should give a large-magnitude slope.

Mishandling a negative coordinate in a difference, e.g. treating $6 - (-2)$ as $4$ instead of $8$ .

Subtracting a negative adds: $6 - (-2) = 6 + 2 = 8$ . Rewrite the double sign before computing.

Using the plain reciprocal for a perpendicular slope — turning $-\tfrac{3}{2}$ into $-\tfrac{2}{3}$ instead of $+\tfrac{2}{3}$ .

Perpendicular slopes are negative reciprocals: flip the fraction AND flip the sign so the product is $-1$ .

For the "other endpoint," averaging the midpoint with the known point instead of extending past it.

The midpoint is between the endpoints, so double it: $x_B = 2x_M - x_A$ . Averaging again lands you halfway, not at the endpoint.

What is the slope of the line passing through the points $(2, 3)$ and $(6, 11)$ ?

What are the coordinates of the midpoint of the segment with endpoints $(-4, 6)$ and $(10, -2)$ ?

What is the distance between the points $(-1, -2)$ and $(3, 1)$ in the standard $(x, y)$ coordinate plane?

Point $M(2, -3)$ is the midpoint of segment $\overline{ST}$ . If $S = (-1, 4)$ , what are the coordinates of point $T$ ?

A line has the equation $y = -\tfrac{4}{3}x + 2$ . What is the slope of a line perpendicular to it?

Triangle $ABC$ has vertices $A(0, 0)$ , $B(6, 0)$ , and $C(6, 8)$ . What is the perimeter of triangle $ABC$ ?

$m = \dfrac{y_2 - y_1}{x_2 - x_1}$

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

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

$m_1 \cdot m_2 = -1$

The xy-Plane Is a Measuring Tool

Slope — Steepness and Direction

The slope $m$ measures how fast $y$ changes as $x$ increases:

$m = \frac{\Delta y}{\Delta x} = \frac{y_2 - y_1}{x_2 - x_1}$

Distance — The Pythagorean Theorem in Disguise

The distance between two points is the length of that hypotenuse:

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

Midpoint — Just an Average

The midpoint of a segment is the average of the endpoints in each direction:

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

Parallel and Perpendicular Lines

Slope also tells you how two lines relate.

Relationship	Slope condition
Parallel	Equal slopes: $m_1 = m_2$
Perpendicular	Negative reciprocals: $m_1 \cdot m_2 = -1$

Putting It on a Graph

When a figure is drawn, read coordinates straight off the grid, then apply the right tool:

How steep / which way? → slope.
How long is a segment? → distance.
Where is the center of a segment? → midpoint.
Do two sides make a right angle? → check that their slopes multiply to $-1$ .

That last test is how the ACT asks you to prove a triangle is right or a quadrilateral is a rectangle without any protractor.

A Reliable Game Plan

Label the points $(x_1, y_1)$ and $(x_2, y_2)$ before plugging in, and keep the order fixed.
Watch every negative coordinate: subtracting a negative adds.
For "other endpoint" problems, use $2x_M - x_1$ rather than re-deriving the average.
To classify lines, compare slopes only — the intercepts are irrelevant.

Common Mistakes to Avoid

Inverting slope as $\tfrac{\Delta x}{\Delta y}$ (run over rise) instead of rise over run.
Forgetting to square inside the distance formula, or taking the root of each term separately.
Using the plain reciprocal — not the negative reciprocal — for a perpendicular slope.
Averaging only the $x$ -coordinates and forgetting the $y$ for a midpoint.