Lesson 4-1, Video 5; Perpendicular Line 2

Summary

This educational script guides viewers through finding the equation of a line perpendicular to a given line. It begins with a line in slope-intercept form, y = -1/7x + 7, and teaches how to derive the perpendicular line's slope, which is the opposite reciprocal of the original. Using the point-slope form, the script demonstrates how to plug in the slope and a given point to find the new line's equation. The lesson then transitions to a unique case where the perpendicular line is to the x-axis, illustrating the concept with a visual example and explaining that vertical lines are represented by x-values, leading to the equation x = 4 for the line passing through (4, -2).

Takeaways

• 📐 The original line is given by the equation y = -1/7x + 7, and its slope is identified as -1/7.
• 🔄 To find a line perpendicular to the original, use the opposite reciprocal of the original slope, resulting in a slope of 7 for the perpendicular line.
• 📍 The perpendicular line must pass through the point (-6, 6), which is provided in the script.
• 📘 The point-slope form of the equation of a line is used, which is y - y1 = m(x - x1), where m is the slope and (x1, y1) is a point on the line.
• 🔢 Plugging in the values for the perpendicular line gives the equation y - 6 = 7(x + 6).
• ✏️ Simplifying the equation from the point-slope form results in the final equation of the perpendicular line.
• 📏 For the second example, a line perpendicular to the x-axis is sought, which means it must be a vertical line.
• 📍 The vertical line should pass through the point (4, -2), as specified in the script.
• 📑 Vertical lines are represented by equations of the form x = constant, using the x-coordinate of the given point.
• 🔑 The final equation for the vertical line is x = 4, which is derived from the x-coordinate of the point (4, -2).

Q & A

What is the slope of the line given by the equation y = -1/7x + 7?

The slope of the line is -1/7.

How do you find the slope of a line perpendicular to a given line?

The slope of a line perpendicular to a given line is the opposite reciprocal of the original line's slope.

What is the slope of the line perpendicular to y = -1/7x + 7?

The slope of the perpendicular line is the opposite reciprocal of -1/7, which is 7.

What is the point-slope form of a line?

The point-slope form of a line is given by the equation y - y1 = m(x - x1), where m is the slope and (x1, y1) is a point on the line.

What is the point-slope form of the line perpendicular to y = -1/7x + 7 that passes through (-6, 6)?

The point-slope form of the line is y - 6 = 7(x + 6).

What is the slope of a line perpendicular to the x-axis?

A line perpendicular to the x-axis is vertical, and its slope is undefined because it has an infinite slope.

How can you represent a vertical line in a coordinate plane?

A vertical line is represented by an equation of the form x = a, where 'a' is the x-coordinate of the line.

What is the equation of the line that is perpendicular to the x-axis and passes through the point (4, -2)?

The equation of the line is x = 4, as it is a vertical line passing through the x-coordinate 4.

Why do horizontal and vertical lines form a 90-degree angle?

Horizontal and vertical lines form a 90-degree angle because they are perpendicular to each other, intersecting at a right angle.

How can you determine if two lines are perpendicular without using slopes?

You can determine if two lines are perpendicular by checking if they intersect at a 90-degree angle, or by visual inspection on a graph.

What is the significance of the opposite reciprocal in finding the slope of a perpendicular line?

The opposite reciprocal is significant because it ensures that the product of the slopes of two perpendicular lines is -1, which is a requirement for perpendicularity.