Use The Given Line And The Point Not On The Line To Answer The Question.What Is The Point On The Line Perpendicular To The Given Line, Passing Through The Given Point That Is Also On The $y$-axis?A. $(-3.6, 0$\]B. $(-2,
Introduction
In this article, we will explore the concept of finding a point on a line that is perpendicular to a given line, passing through a given point that lies on the y-axis. This problem involves understanding the properties of lines, particularly the concept of perpendicularity and the equation of a line.
Understanding the Problem
The problem statement asks us to find the point on the line that is perpendicular to the given line, passing through the given point that lies on the y-axis. To solve this problem, we need to understand the following key concepts:
- Perpendicular lines: Two lines are perpendicular if they intersect at a right angle (90 degrees).
- Equation of a line: The equation of a line can be written in the form y = mx + b, where m is the slope of the line and b is the y-intercept.
- Slope: The slope of a line is a measure of how steep the line is. It is calculated as the ratio of the vertical change (rise) to the horizontal change (run).
Step 1: Identify the Given Line and Point
The given line is not explicitly stated in the problem, but we can assume it to be a line with a given equation. Let's assume the given line is y = 2x + 3. The given point that lies on the y-axis is (0, 4).
Step 2: Find the Slope of the Given Line
To find the slope of the given line, we can use the equation of the line. The slope of the line y = 2x + 3 is 2.
Step 3: Find the Slope of the Perpendicular Line
Since the perpendicular line is perpendicular to the given line, its slope is the negative reciprocal of the slope of the given line. The negative reciprocal of 2 is -1/2.
Step 4: Find the Equation of the Perpendicular Line
Using the point-slope form of a line, we can write the equation of the perpendicular line as y - 4 = (-1/2)(x - 0). Simplifying this equation, we get y = (-1/2)x + 4.
Step 5: Find the Point on the Line
To find the point on the line that is perpendicular to the given line, passing through the given point that lies on the y-axis, we need to find the intersection point of the two lines. We can do this by equating the two equations and solving for x.
Equating the two equations, we get:
2x + 3 = (-1/2)x + 4
Simplifying this equation, we get:
(5/2)x = 1
x = 2/5
Substituting this value of x into the equation of the given line, we get:
y = 2(2/5) + 3
y = 16/5
Therefore, the point on the line that is perpendicular to the given line, passing through the given point that lies on the y-axis is (2/5, 16/5).
Conclusion
In this article, we have solved the problem of finding the point on the line that is perpendicular to the given line, passing through the given point that lies on the y-axis. We have used the concept of perpendicular lines, the equation of a line, and the slope of a line to find the solution. The point on the line that is perpendicular to the given line, passing through the given point that lies on the y-axis is (2/5, 16/5).
Answer
The correct answer is (2/5, 16/5).
Comparison with Options
Let's compare our answer with the given options:
A. (-3.6, 0) B. (-2, 0)
Our answer (2/5, 16/5) is different from both options A and B.
Final Thoughts
Q: What is the concept of perpendicular lines?
A: Perpendicular lines are two lines that intersect at a right angle (90 degrees). This means that if you draw a line from one point on one line to a point on the other line, the angle between the two lines is 90 degrees.
Q: How do you find the slope of a line?
A: The slope of a line is a measure of how steep the line is. It is calculated as the ratio of the vertical change (rise) to the horizontal change (run). The slope of a line can be found using the equation of the line.
Q: What is the equation of a line?
A: The equation of a line can be written in the form y = mx + b, where m is the slope of the line and b is the y-intercept.
Q: How do you find the equation of a perpendicular line?
A: To find the equation of a perpendicular line, you need to find the slope of the perpendicular line, which is the negative reciprocal of the slope of the original line. Then, you can use the point-slope form of a line to write the equation of the perpendicular line.
Q: What is the point-slope form of a line?
A: The point-slope form of a line is a way of writing the equation of a line that passes through a given point and has a given slope. It is written in the form y - y1 = m(x - x1), where (x1, y1) is the given point and m is the slope.
Q: How do you find the intersection point of two lines?
A: To find the intersection point of two lines, you need to set the two equations equal to each other and solve for x. Then, you can substitute the value of x into one of the equations to find the value of y.
Q: What is the significance of the y-axis in this problem?
A: The y-axis is significant in this problem because the given point lies on the y-axis. This means that the x-coordinate of the given point is 0.
Q: How do you find the point on the line that is perpendicular to the given line?
A: To find the point on the line that is perpendicular to the given line, you need to find the intersection point of the two lines. This can be done by setting the two equations equal to each other and solving for x. Then, you can substitute the value of x into one of the equations to find the value of y.
Q: What is the final answer to the problem?
A: The final answer to the problem is (2/5, 16/5).
Q: How do you compare the final answer with the given options?
A: To compare the final answer with the given options, you need to check if the final answer matches any of the options. In this case, the final answer (2/5, 16/5) does not match either option A or option B.
Q: What are some common mistakes to avoid when solving this problem?
A: Some common mistakes to avoid when solving this problem include:
- Not understanding the concept of perpendicular lines
- Not finding the slope of the perpendicular line correctly
- Not using the point-slope form of a line correctly
- Not finding the intersection point of the two lines correctly
Q: How do you ensure that you are solving the problem correctly?
A: To ensure that you are solving the problem correctly, you need to:
- Read the problem carefully and understand what is being asked
- Use the correct formulas and equations
- Check your work carefully to avoid mistakes
- Use a calculator or computer program to check your answer if necessary
Q: What are some real-world applications of this problem?
A: Some real-world applications of this problem include:
- Finding the intersection point of two lines in a coordinate plane
- Determining the slope of a line in a coordinate plane
- Finding the equation of a line in a coordinate plane
- Solving systems of linear equations in a coordinate plane