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Introduction
Perpendicular Lines helps you connect a mathematical rule to a result you can explain and check. Use a negative reciprocal slope for nonvertical, nonhorizontal lines. Handle a horizontal-vertical pair separately. This lesson belongs to coordinate geometry and graphing and builds toward independent Algebra I problem solving.
Alignment uses 19 TAC §111.39(c). Review lessons sample their listed standards. Readiness and supporting labels describe STAAR emphasis, not whether a topic is required.
Learning objectives
- Identify the quantities and representations used in perpendicular lines.
- Apply the lesson's rule to solve a numerical or contextual problem.
- Explain each mathematical step using the correct vocabulary.
- Verify an answer and correct a likely error.
Prerequisite knowledge
Divide fractions and write point-slope equations.
Previous lesson: Parallel Lines
Key mathematical vocabulary
- slope
- Vertical change divided by nonzero horizontal change.
- rate
- A ratio measured in units per unit.
- expression
- A mathematical phrase representing a value.
- solution
- A value making the original conditions true.
- input
- A value supplied to a rule.
Formulas and rules
- y=mx+b
- m=(y₂−y₁)/(x₂−x₁), x₂≠x₁
- y−y₁=m(x−x₁)
Perpendicular lines through a point
Use a negative reciprocal slope for nonvertical, nonhorizontal lines. Handle a horizontal-vertical pair separately.
Perpendicular lines meet at a right angle. For two finite nonzero slopes, their product is −1, so one slope is the negative reciprocal of the other. Use the new slope and the supplied point to write the desired line.
The negative reciprocal changes both the sign and numerator-denominator order. For example, the negative reciprocal of 2/3 is −3/2. A horizontal line and a vertical line are also perpendicular; they form the special case where the finite-slope product rule cannot be used.
A method you can explain
- Read the question and identify the known information, unknown quantities, and any domain restrictions. Select a suitable representation and tool.
- Apply the relationship explained above. The given slope is 12.
- Write each calculation or logical inference on a separate line. Keep units and signs visible. When two methods are possible, compare which one uses the given information most directly.
- Check the result in the ORIGINAL conditions, not merely the last simplified line. The new line passes through the specified point. Check the required slope relationship separately.
- Interpret what the answer means. Decide whether its size, sign, and precision make sense; identify a restriction or assumption that would change the conclusion.
Five worked examples
Example 1 · Beginner
Write a line through (6, 18) perpendicular to y=12x+90.
- The given slope is 12.
- A perpendicular slope is the negative reciprocal: −1/12; the product is −1.
- Use point-slope form with the given point: y − 18 = −(1/12)(x − 6).
Answer: y − 18 = −(1/12)(x − 6)
Verify: The new line passes through the specified point. Check the required slope relationship separately.
Avoid this mistake: Changing the sign without taking the reciprocal.
Example 2 · Basic
Write a line through (6, 18) perpendicular to y=13x+96.
- The given slope is 13.
- A perpendicular slope is the negative reciprocal: −1/13; the product is −1.
- Use point-slope form with the given point: y − 18 = −(1/13)(x − 6).
Answer: y − 18 = −(1/13)(x − 6)
Verify: The new line passes through the specified point. Check the required slope relationship separately.
Avoid this mistake: Taking the reciprocal without changing the sign.
Example 3 · Intermediate
Write a line through (6, 18) perpendicular to y=14x+102.
- The given slope is 14.
- A perpendicular slope is the negative reciprocal: −1/14; the product is −1.
- Use point-slope form with the given point: y − 18 = −(1/14)(x − 6).
Answer: y − 18 = −(1/14)(x − 6)
Verify: The new line passes through the specified point. Check the required slope relationship separately.
Avoid this mistake: Applying the product-of-slopes rule to a vertical line.
Example 4 · Advanced
Write a line through (6, 19) perpendicular to y=2x+31.
- The given slope is 2.
- A perpendicular slope is the negative reciprocal: −1/2; the product is −1.
- Use point-slope form with the given point: y − 19 = −(1/2)(x − 6).
Answer: y − 19 = −(1/2)(x − 6)
Verify: The new line passes through the specified point. Check the required slope relationship separately.
Avoid this mistake: Changing the sign without taking the reciprocal.
Example 5 · Challenge
Write a line through (6, 19) perpendicular to y=3x+37.
- The given slope is 3.
- A perpendicular slope is the negative reciprocal: −1/3; the product is −1.
- Use point-slope form with the given point: y − 19 = −(1/3)(x − 6).
Answer: y − 19 = −(1/3)(x − 6)
Verify: The new line passes through the specified point. Check the required slope relationship separately.
Avoid this mistake: Taking the reciprocal without changing the sign.
Common mistakes and corrections
- Changing the sign without taking the reciprocal.
- Taking the reciprocal without changing the sign.
- Applying the product-of-slopes rule to a vertical line.
Real-world applications
- Plan paths that meet at right angles on a coordinate map. Define the input and its units, apply the mathematical relationship, and check that the result fits the setting.
- Construct a perpendicular boundary through a specified survey point. Define the input and its units, apply the mathematical relationship, and check that the result fits the setting.
Topic summary
Use a negative reciprocal slope for nonvertical, nonhorizontal lines. Handle a horizontal-vertical pair separately.
Perpendicular lines meet at a right angle. For two finite nonzero slopes, their product is −1, so one slope is the negative reciprocal of the other. Use the new slope and the supplied point to write the desired line.
Frequently asked questions
What should I know before perpendicular lines?
Divide fractions and write point-slope equations.
How can I check my work?
The new line passes through the specified point. Check the required slope relationship separately. A numerical check can detect an error; when claiming an identity for all inputs, also explain the algebraic property that makes it true.
Which mistakes should I watch for?
Changing the sign without taking the reciprocal. Taking the reciprocal without changing the sign. Applying the product-of-slopes rule to a vertical line.
Can I use another valid method?
Yes. Explain your assumptions, show enough intermediate work to justify the result, and verify it in the original problem. Equivalent expressions and valid alternative methods receive credit.
Mathematical process practice · A.1A–A.1G
- A.1A: Plan paths that meet at right angles on a coordinate map.
- A.1B: Plan, solve, check, and interpret the written response.
- A.1C: Select paper, a calculator, a table, or a graph; explain the choice.
- A.1D: Connect symbols with a verbal explanation and another representation.
- A.1E: Organize the solution with a table, diagram, or labeled equations.
- A.1F: Explain how the representation and the algebraic rule express the same relationship.
- A.1G: Justify each operation and the reasonableness of the conclusion.
Quick revision notes
- Use a negative reciprocal slope for nonvertical, nonhorizontal lines. Handle a horizontal-vertical pair separately.
- Changing the sign without taking the reciprocal.
- Taking the reciprocal without changing the sign.
- Applying the product-of-slopes rule to a vertical line.
Practice and explain
Try all four multiple-choice questions, then show your reasoning in the written response. Objective scores and AI feedback assess different skills.
Continue learning
Original Cambridge Veritas practice. These are not official STAAR items. Cambridge Veritas is not affiliated with or endorsed by TEA.
Official Algebra I TEKS (§111.39) · TEA assessed curriculum · STAAR blueprint · TEA instructional materials