ACT® · Math: Geometry — Coordinate Geometry and Conic Sections
Finding a Point From Parallel or Perpendicular Conditions
A parallel or perpendicular condition pins down a slope, and a slope equation with one unknown coordinate is just an equation you solve. Set the slope formula equal to the required value and isolate the missing number.
A single condition like 'parallel to this line' or 'perpendicular to this side' is really a hidden equation. Parallel lines share the same slope. Perpendicular lines have slopes that multiply to -1, so one slope is the negative reciprocal of the other. Once you know the slope a segment must have, and you know one point on it plus one coordinate of the second point, the slope formula (y2-y1)/(x2-x1) has only one unknown left. Solve for it like any algebra equation.
The mechanism is always the same three steps: figure out the target slope from the parallel or perpendicular rule, write the slope formula between your two points, then solve. For perpendicular problems, flip the given slope and change its sign before you set up the equation. For rectangles, a faster route uses vectors instead of slopes: opposite sides are parallel and adjacent sides are perpendicular, so the fourth vertex is just the sum of the two edge vectors drawn from a shared corner. This works because a rectangle's fourth corner closes the parallelogram, and vector addition does that automatically without ever computing a slope.
This topic sits right after the basic slope, midpoint, and distance formulas, and it is the natural next step once those are solid. It also previews conic work later, since recognizing perpendicularity by slope product is the same skill used to check tangent lines and axes of symmetry in circles and parabolas.
Key terms
- slope
- The ratio of vertical change to horizontal change between two points on a line.
- negative reciprocal
- The value you get by flipping a fraction and changing its sign, used for perpendicular slopes.
- vector sum
- Adding the horizontal and vertical components of two vectors to get a new point or vector.
- parallelogram closure
- The idea that the fourth vertex of a parallelogram completes the figure so opposite sides stay equal and parallel.
Watch it explained
- How it worksHow do you turn the words 'these two sides meet at a right angle' into an equation you can solve?
- Worked exampleFind k so that the line through (1, 3) and (k, 9) runs parallel to y = 2x - 7.
- Worked exampleFind the fourth vertex of a rectangle whose other three corners sit at (0, 0), (6, 2) and (-1, 3).
Check yourself
End of topic.