Line \[$ S \$\] Has An Equation Of $ Y = \frac{1}{3}x - 5 $. Line \[$ T \$\] Is Perpendicular To Line \[$ S \$\] And Passes Through The Point \[$(-2, 9)\$\].What Is The Equation Of Line \[$ T \$\]?Write

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Introduction

In mathematics, the concept of perpendicular lines is crucial in understanding various geometric and algebraic relationships. When two lines are perpendicular, their slopes are negative reciprocals of each other. In this article, we will explore how to find the equation of a line that is perpendicular to a given line and passes through a specific point.

Understanding the Problem

We are given a line { s $}$ with the equation $ y = \frac{1}{3}x - 5 $. Our goal is to find the equation of a line { t $}$ that is perpendicular to line { s $}$ and passes through the point {(-2, 9)$}$.

Recalling the Concept of Slope

The slope of a line is a measure of how steep it is. It is calculated as the ratio of the vertical change (rise) to the horizontal change (run) between two points on the line. The slope of line { s $}$ is 13\frac{1}{3}.

Finding the Slope of the Perpendicular Line

Since line { t $}$ is perpendicular to line { s $}$, its slope will be the negative reciprocal of the slope of line { s $}$. The negative reciprocal of 13\frac{1}{3} is −3-3.

Using the Point-Slope Form

The point-slope form of a line is given by the equation y−y1=m(x−x1)y - y_1 = m(x - x_1), where mm is the slope and (x1,y1)(x_1, y_1) is a point on the line. We can use this form to find the equation of line { t $}$.

Substituting the Values

We know that the slope of line { t $}$ is −3-3 and it passes through the point (−2,9)(-2, 9). Substituting these values into the point-slope form, we get:

y−9=−3(x−(−2))y - 9 = -3(x - (-2))

Simplifying the Equation

Expanding the equation, we get:

y−9=−3(x+2)y - 9 = -3(x + 2)

Distributing the −3-3, we get:

y−9=−3x−6y - 9 = -3x - 6

Adding 99 to both sides, we get:

y=−3x+3y = -3x + 3

Conclusion

In this article, we found the equation of a line that is perpendicular to a given line and passes through a specific point. We used the concept of slope and the point-slope form to derive the equation of the perpendicular line. The equation of line { t $}$ is y=−3x+3y = -3x + 3.

Example Use Cases

  1. Architecture: When designing buildings, architects need to ensure that the lines of the building's facade are perpendicular to each other. This is crucial for maintaining the structural integrity of the building.
  2. Engineering: In engineering, perpendicular lines are used to design and build various structures such as bridges, roads, and buildings.
  3. Computer Graphics: In computer graphics, perpendicular lines are used to create 3D models and animations.

Tips and Tricks

  1. Use the Negative Reciprocal: When finding the slope of a perpendicular line, remember to use the negative reciprocal of the slope of the original line.
  2. Use the Point-Slope Form: The point-slope form is a useful tool for finding the equation of a line that passes through a specific point.
  3. Simplify the Equation: Make sure to simplify the equation by combining like terms and eliminating any unnecessary variables.

Conclusion

Q: What is the concept of perpendicular lines?

A: Perpendicular lines are lines that intersect at a 90-degree angle. In other words, they are lines that are at right angles to each other.

Q: How do I find the slope of a perpendicular line?

A: To find the slope of a perpendicular line, you need to find the negative reciprocal of the slope of the original line. For example, if the slope of the original line is 2, the slope of the perpendicular line would be -1/2.

Q: What is the point-slope form of a line?

A: 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.

Q: How do I use the point-slope form to find the equation of a perpendicular line?

A: To use the point-slope form to find the equation of a perpendicular line, you need to substitute the values of the slope and the point into the equation. For example, if the slope of the perpendicular line is -3 and it passes through the point (-2, 9), you would substitute these values into the equation y - 9 = -3(x - (-2)).

Q: What is the equation of a line that is perpendicular to the line y = 1/3x - 5 and passes through the point (-2, 9)?

A: To find the equation of a line that is perpendicular to the line y = 1/3x - 5 and passes through the point (-2, 9), you need to find the slope of the perpendicular line, which is the negative reciprocal of the slope of the original line. The slope of the original line is 1/3, so the slope of the perpendicular line is -3. You can then use the point-slope form to find the equation of the perpendicular line.

Q: How do I simplify the equation of a line?

A: To simplify the equation of a line, you need to combine like terms and eliminate any unnecessary variables. For example, if the equation of a line is y - 9 = -3(x + 2), you can simplify it by adding 9 to both sides to get y = -3x + 3.

Q: What are some real-world applications of perpendicular lines?

A: Perpendicular lines have many real-world applications, including architecture, engineering, and computer graphics. In architecture, perpendicular lines are used to design buildings and ensure that the lines of the building's facade are perpendicular to each other. In engineering, perpendicular lines are used to design and build various structures such as bridges, roads, and buildings. In computer graphics, perpendicular lines are used to create 3D models and animations.

Q: How do I find the equation of a line that is perpendicular to a given line and passes through a specific point?

A: To find the equation of a line that is perpendicular to a given line and passes through a specific point, you need to follow these steps:

  1. Find the slope of the given line.
  2. Find the negative reciprocal of the slope of the given line.
  3. Use the point-slope form to find the equation of the perpendicular line.
  4. Simplify the equation by combining like terms and eliminating any unnecessary variables.

Q: What is the equation of a line that is perpendicular to the line y = 2x + 3 and passes through the point (1, 5)?

A: To find the equation of a line that is perpendicular to the line y = 2x + 3 and passes through the point (1, 5), you need to follow the steps outlined above. The slope of the given line is 2, so the slope of the perpendicular line is -1/2. You can then use the point-slope form to find the equation of the perpendicular line.

Q: How do I use the equation of a line to solve problems?

A: To use the equation of a line to solve problems, you need to substitute the values of the variables into the equation and solve for the unknown variable. For example, if the equation of a line is y = -3x + 3 and you know that the value of y is 6, you can substitute this value into the equation and solve for x.

Q: What are some common mistakes to avoid when working with perpendicular lines?

A: Some common mistakes to avoid when working with perpendicular lines include:

  • Not finding the negative reciprocal of the slope of the given line.
  • Not using the point-slope form to find the equation of the perpendicular line.
  • Not simplifying the equation by combining like terms and eliminating any unnecessary variables.
  • Not substituting the values of the variables into the equation and solving for the unknown variable.