Coordinate proofs use the distance, midpoint and slope formulas to verify geometric facts on the plane. Two lines are parallel when their slopes are equal; they are perpendicular when their slopes are negative reciprocals (their product is −1). You can show a figure is a parallelogram by proving opposite sides have equal slopes, a right angle by showing two sides have negative-reciprocal slopes, or an isosceles triangle by showing two sides have equal length via the distance formula. The SAT most often tests the parallel and perpendicular slope rules — given a line’s slope, state the slope of a line parallel or perpendicular to it.
✅ Solved examples
1. A line has slope 3. What is the slope of a parallel line?
Parallel lines have equal slopes: 3.
2. A line has slope 3. What is the slope of a perpendicular line?
Negative reciprocal: −1/3.
3. A line has slope −2. Slope of a perpendicular line?
Negative reciprocal: 1/2.
4. Two lines have slopes 4 and 4. Are they parallel or perpendicular?
Equal slopes, so parallel.
✏️ Practice — try these, take hints as needed
1. A line has slope 5. Slope of a parallel line?
Parallel = equal slopes.
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—
5.
2. A line has slope 2. Slope of a perpendicular line?
Negative reciprocal.
−1/2.
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−1/2.
3. A line has slope −4. Slope of a perpendicular line?
Flip and change sign.
1/4.
—
1/4.
4. Two lines have slopes 3 and −1/3. Parallel or perpendicular?
Product = −1?
3 × (−1/3) = −1.
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Perpendicular.
5. A line has slope 6. Slope of a line parallel to it?
Equal slopes.
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6.
📝 Topic test — 8 questions
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