A reliable method
Draw a quick sketch to identify the requested quantity. For a perpendicular line with nonzero slope m, use −1/m and check that the product is −1.
The common trap
A midpoint is not the difference of endpoints. A line parallel to another has the same slope but does not necessarily share the same intercept.
Make the connection
A correct equation should reproduce the given point. A distance should be nonnegative and at least as large as either coordinate difference in magnitude.
Worked example: choose the requested quantity
A line has slope 3/3. What is the slope of a perpendicular line? For two nonvertical perpendicular lines, the slopes multiply to −1. Take the reciprocal and change its sign. The requested value is -1.
Contrast and diagnose an error
What is the x-coordinate of the midpoint of (3, −2) and (8, 8)? A midpoint lies halfway between the endpoint coordinates. Average the x-values; the y-values do not affect this requested coordinate. The requested value is 11/2. A tempting answer is 3: Average the y coordinates.
Transfer practice and check
Use the labeled coordinate points in the graph. What is the straight-line distance between them? Coordinate points A(-1,3) and B(8,15); connecting segment. Sketch not to scale. Try it before reading on. Check: 15. The horizontal change is 9 and the vertical change is 12. Use the distance formula, recognizing a scaled 3–4–5 right triangle.
A line has slope 3/3. What is the slope of a perpendicular line?
- For two nonvertical perpendicular lines, the slopes multiply to −1.
- Take the reciprocal and change its sign.
- The requested value is -1.
Compare the other answer choices
Treat perpendicular as horizontal regardless of the original slope.
Take the reciprocal but omit the negative sign.
Divide the reciprocal slope by two as if averaging directions.
Try it, then explain it.
Complete a short set without looking at the example. After each answer, explain which operation preserved the relationship and why the result makes sense.
Sources & authorship
Original Slope explanations and examples. References support the concepts; they are not the source of copied questions. Independent educator review is pending.
ACT Mathematics content