Perpendicular Lines and Point-Slope Form: Unlocking the Equation

Algebra 2 Grades High School 4:10 Video
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Lesson Description

Learn how to write the equation of a line in point-slope form when given a point and a perpendicular line. Master the relationship between slopes of perpendicular lines and apply it to solve problems.

Video Resource

Point Slope Form Equation | Given Point and Perpendicular Line

Kevinmathscience

Duration: 4:10
Watch on YouTube

Key Concepts

  • Point-slope form of a linear equation
  • Slope of a line
  • Perpendicular lines and their slopes
  • Negative reciprocal
  • Slope-intercept form (y = mx + b)

Learning Objectives

  • Students will be able to determine the slope of a line perpendicular to a given line.
  • Students will be able to write the equation of a line in point-slope form given a point and a line perpendicular to it.
  • Students will be able to apply the concept of negative reciprocal slopes to solve problems.
  • Students will be able to recognize slope-intercept form.

Educator Instructions

  • Introduction (5 mins)
    Begin by reviewing the point-slope form of a linear equation: y - y1 = m(x - x1). Briefly discuss the meaning of 'm' (slope) and (x1, y1) (a point on the line). Remind students about slope intercept form.
  • Perpendicular Lines (10 mins)
    Explain the relationship between the slopes of perpendicular lines (they are negative reciprocals of each other). Emphasize that the product of the slopes of two perpendicular lines is -1. Provide examples of finding the slope of a perpendicular line when given the slope of another line (e.g., if the slope is 2, the perpendicular slope is -1/2).
  • Video Example 1 (10 mins)
    Play the first example from the Kevinmathscience video. Pause at key points to explain the steps involved: identifying the slope of the given line, finding the slope of the perpendicular line, and plugging the point and the new slope into the point-slope form. Ensure students understand each step.
  • Video Example 2 (10 mins)
    Play the second example from the video. Encourage students to predict the next step before it's shown. Reinforce the concepts of identifying the slope and using the given point. Discuss simplification of the equation.
  • Practice Problems (15 mins)
    Provide students with practice problems where they need to find the equation of a line in point-slope form given a point and a line perpendicular to it. Walk around the classroom to offer assistance and answer questions.
  • Wrap-up (5 mins)
    Summarize the key concepts of the lesson. Answer any remaining questions. Preview the next lesson topic.

Interactive Exercises

  • Slope Matching Game
    Provide students with a list of slopes. They need to match each slope with its negative reciprocal.
  • Equation Creation Challenge
    Give students a point and an equation of a line. Ask them to find the equation of the perpendicular line through the given point.

Discussion Questions

  • Why is the concept of negative reciprocal important when dealing with perpendicular lines?
  • How does point-slope form help us write equations of lines when we don't know the y-intercept?
  • Can you think of real-world examples where perpendicular lines are important?

Skills Developed

  • Algebraic manipulation
  • Problem-solving
  • Critical thinking
  • Application of mathematical concepts

Multiple Choice Questions

Question 1:

What is the point-slope form of a linear equation?

Correct Answer: y - y1 = m(x - x1)

Question 2:

If a line has a slope of 3, what is the slope of a line perpendicular to it?

Correct Answer: -1/3

Question 3:

The slope of a line is -2/5. What is the slope of a line perpendicular to it?

Correct Answer: 5/2

Question 4:

A line passes through the point (1, 4) and is perpendicular to a line with a slope of 2. What is the equation of the line in point slope form?

Correct Answer: y - 4 = -1/2(x - 1)

Question 5:

What does 'm' represent in the point-slope form equation?

Correct Answer: slope

Question 6:

A line is perpendicular to y = (1/4)x + 2. What is the value of the slope of the perpendicular line?

Correct Answer: -4

Question 7:

What is the negative reciprocal of -5?

Correct Answer: 1/5

Question 8:

A line passes through (2, -3) and is perpendicular to a line with slope 1. What is the equation in point-slope form?

Correct Answer: y + 3 = -1(x - 2)

Question 9:

Which of the following represents slope-intercept form?

Correct Answer: y = mx + b

Question 10:

A line perpendicular to y + 1 = 3(x-5) also passes through point (0,0). What is the equation of the perpendicular line in point-slope form?

Correct Answer: y = -1/3(x)

Fill in the Blank Questions

Question 1:

The point-slope form of a linear equation is y - y1 = m(x - _____).

Correct Answer: x1

Question 2:

Perpendicular lines intersect at a _____ angle.

Correct Answer: 90

Question 3:

If a line has a slope of 4, the slope of a line perpendicular to it is _____.

Correct Answer: -1/4

Question 4:

The negative reciprocal of 2/3 is _____.

Correct Answer: -3/2

Question 5:

To find the equation of a perpendicular line you must use its _____.

Correct Answer: slope

Question 6:

A line goes through point (5, 2) and has slope of -1. The equation for point slope form is: y - _____ = -1(x - 5).

Correct Answer: 2

Question 7:

In the equation y = mx + b, 'b' stands for the _____.

Correct Answer: y-intercept

Question 8:

Two parallel lines will never _____.

Correct Answer: intersect

Question 9:

A line with the equation y - 1 = 2(x + 4) passes through point (-4, _____).

Correct Answer: 1

Question 10:

The product of the slopes of two perpendicular lines is always _____.

Correct Answer: -1

Educational Standards

A2.A.1:Represent and solve mathematical and real-world problems using nonlinear equations, systems of linear equations, and systems of linear inequalities; interpret the solutions in the original context. A2.A.2:Generate and evaluate equivalent algebraic expressions and equations using various strategies. A2.F.1:Understand functions as descriptions of covariation (how related quantities vary together).

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