answer:
Certainly! I'd be happy to help you with your question.
To construct a line through the point perpendicular to a given line, we can use the following steps:
Step 1: Identify the given line
First, we need to identify the line that we want to find the perpendicular line through. Let's say the given line is:
y = 2x + 3
Step 2: Find the slope of the given line
To find the slope of the line, we can use the formula:
slope = (rise / run) = (y - y1) / (x - x1)
where (x1, y1) and (x2, y2) are two points on the line.
In this case, we can choose any two points on the line, such as (1, 5) and (4, 11).
slope = (11 - 5) / (4 - 1) = 6 / 3 = 2
So, the slope of the given line is 2.
Step 3: Find the point where the perpendicular line intersects the given line
To find the point where the perpendicular line intersects the given line, we can use the formula:
point = (x, y) = (x1 + slope * (x2 - x1), y1 + slope * (y2 - y1))
where (x1, y1) and (x2, y2) are two points on the given line.
In this case, we can choose the points as before:
point = (1 + 2 * (4 - 1), 5 + 2 * (11 - 5))
point = (5, 13)
So, the point where the perpendicular line intersects the given line is (5, 13).
Step 4: Construct the perpendicular line
Now that we have the point where the perpendicular line intersects the given line, we can construct the perpendicular line. We can do this by drawing a line that passes through the point and is perpendicular to the given line.
To check that the line is perpendicular, we can use the fact that the dot product of the two lines must be zero:
dot product = (y - y1) * (x - x1) - (x - x1) * (y - y1)
where (x1, y1) and (x2, y2) are two points on the lines.
In this case, we can use the points (1, 5) and (5, 13):
dot product = (13 - 5) * (5 - 1) - (5 - 1) * (13 - 5)
dot product = 8 * 4 - 5 * 8
dot product = 32 - 40
dot product = -8
Since the dot product is negative, we know that the two lines are perpendicular.
Therefore, the perpendicular line to the given line y = 2x + 3 is the line:
y = -2x + 13
This line passes through the point (5, 13) and is perpendicular to the given line.
And that's it!
To support this answer, I would recommend the following three authoritative reference titles:
1. "Calculus" by Michael Spivak - This comprehensive calculus textbook covers the basics of calculus, including the concept of slopes and perpendicular lines.
2. "Linear Algebra and Its Applications" by Gilbert Strang - This linear algebra textbook covers the basics of linear algebra, including the concept of dot products and perpendicular lines.
3. "Geometry: Seeing, Doing, Understanding" by Harold R. Jacobs - This geometry textbook covers the basics of geometry, including the concept of perpendicular lines and their properties.
I hope this helps! Let me know if you have any other questions.