Decision and conditions
Recover geometric information through midpoint, distance, reflection and perpendicularity. For nonvertical perpendicular lines, slopes have product minus one; distance is nonnegative.
Worked model
A line is perpendicular to y = 2x + 3 and passes through (6, 6). What is its y-intercept? The perpendicular slope is −1/2. In y = −x/2 + b, substitute (6,6). b = 6 + 3 = 9. Before calculating: Use the negative reciprocal slope, then substitute the stated point.
Contrast: a different decision
The midpoint of A=(3,2) and B=(b,8) is (7,5). What is b? (3+b)/2 = 7. Double the midpoint coordinate, then subtract the known endpoint. b = 14 − 3 = 11. Before calculating: The midpoint coordinate is the average of the endpoint coordinates.
Transfer with solution
What is the distance between (3,−3) and (9,5)? The coordinate differences are 6 and 8. Distance squared is 36 + 64 = 100. The nonnegative distance is 10. Before calculating: Find the horizontal and vertical differences before using the distance formula.
Check the boundary
A reflected point changes only the coordinate perpendicular to the mirror axis. Point P=(3,−5) is reflected across the y-axis to Q. Quantity A: distance OP. Quantity B: distance OQ, where O is the origin. Q = (−3,−5). The squared distances use the same two coordinate squares. Reflection preserves distance to the origin, so the quantities are equal.
A line is perpendicular to y = 2x + 3 and passes through (6, 6). What is its y-intercept?
- The perpendicular slope is −1/2.
- In y = −x/2 + b, substitute (6,6).
- b = 6 + 3 = 9.
Compare the other answer choices
Uses positive reciprocal slope 1/2 instead of the negative reciprocal.
Reports the given point’s y-coordinate as the intercept.
Uses slope −1 instead of −1/2, adding the full x-coordinate to y.
Uses the original slope 2: b = 6 − 2(6) = -6.
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.
GRE Quantitative Reasoning overview