Coordinate geometry on the ACT

Coordinate geometry on the ACT is mostly five tools: slope, slope-intercept form, the rules for parallel and perpendicular lines, the midpoint, and the distance formula. None of them require insight in the moment, which is what makes this topic so valuable. Once they are automatic, coordinate questions become some of the quickest points in the section.

Slope is the engine

Slope is the change in y divided by the change in x, better known as rise over run. Given two points, subtract the y values, subtract the x values in the same order, and divide. For (2, 3) and (6, 11), the change in y is 8 and the change in x is 4, so the slope is 2.

The phrase "in the same order" is doing heavy lifting there. You may start with either point, but whichever one you subtract from first on top, you must subtract from first on the bottom. Flipping only one of them gives negative 2, and both 2 and negative 2 will be in the choices.

A visual check keeps you honest. A line going up as you move right has a positive slope, a line going down has a negative slope, a horizontal line has a slope of 0, and a vertical line has an undefined slope.

Slope-intercept form

The workhorse equation is y = mx + b, where m is the slope and b is the y-intercept. To write the equation of the line through (2, 3) and (6, 11), start with the slope of 2 from above, then plug one point in: 3 equals 2 times 2 plus b, so b is negative 1 and the line is y = 2x - 1.

Always test your equation on the other point. Plugging in x equals 6 gives 12 minus 1, which is 11, matching exactly. That ten-second check catches arithmetic slips before they become wrong answers. If rearranging equations into this form feels shaky, working with linear equations is the place to shore that up.

Parallel and perpendicular

Parallel lines have equal slopes. Perpendicular lines have slopes that are negative reciprocals, which means their product is negative 1. Flip the fraction, flip the sign.

• The line parallel to y = -3x + 5 through (1, 4) has a slope of negative 3, so 4 equals negative 3 plus b, making b equal 7. The line is y = -3x + 7.
• A slope of two thirds is perpendicular to a slope of negative three halves, since their product is negative 1.
• A slope of 4 is perpendicular to a slope of negative one fourth. Whole numbers still get flipped.

Midpoint is just an average

The midpoint of a segment is the average of the two x values paired with the average of the two y values. For (negative 4, 7) and (10, 1), the x value is negative 4 plus 10 over 2, which is 3, and the y value is 7 plus 1 over 2, which is 4. The midpoint is (3, 4).

A harder version gives you the midpoint and one endpoint. Work the average backwards. If the midpoint is (5, negative 2) and one endpoint is (1, 4), then 1 plus the missing x equals 10, so the missing x is 9, and 4 plus the missing y equals negative 4, so the missing y is negative 8. The other endpoint is (9, negative 8).

The distance formula is the Pythagorean theorem

You can memorize the distance formula, but it is easier to rebuild it. Two points on the grid form the corners of a right triangle: the horizontal gap is one leg, the vertical gap is the other, and the distance is the hypotenuse. From (1, 2) to (7, 10), the horizontal gap is 6 and the vertical gap is 8, so 36 plus 64 is 100 and the distance is the square root of 100, which is 10. When a distance does not come out clean, leave it as a square root and match it to the choices. The same right triangle thinking is covered from the other direction in how to solve ACT triangle problems.

Traps to watch for

• Sign errors when subtracting negatives. Rewriting 7 minus negative 3 as 7 plus 3 before computing prevents most of them.
• Reversing the coordinates. A point is always (x, y), so a y-intercept of 4 is the point (0, 4), not (4, 0).
• Forgetting to flip the sign on a perpendicular slope. The reciprocal-only answer is almost always offered.

A quick self-check

Find the slope through (negative 2, 5) and (4, negative 7). Then find the midpoint of those same two points. Then write the equation of the line perpendicular to y = 2x + 9 passing through (0, 3). The answers are a slope of negative 2, since negative 12 over 6 is negative 2; a midpoint of (1, negative 1); and y = -0.5x + 3, since the perpendicular slope is negative one half and the point given is already the y-intercept.

Every one of these rewards repetition over cleverness, so a short set of coordinate questions on thirty-six most days will make them automatic. From here, the natural extensions are circles on the coordinate plane and the broader picture in how much geometry is on the ACT.

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