Question

which step is the same in the construction of a perpendicular line through a point on a line and the construction of a perpendicular line through a point off a line?

Answer 1

A line is said to be perpendicular to another line if the two lines intersect at a right angle. Explicitly, a first line is perpendicular to a second line if (1) the two lines meet; and (2) at the point of intersection the straight angle on one side of the first line is cut by the second line into two congruent angles.

Answer 2

We want to find a perpendicular line to a given line, such that it passes through a point (and we want to separate the case where the point is on the line and when the point is not in the line, we will see that the construction is exactly the same).

Well, let's define the first line:

f(x) = a×x + b

where a is the slope and b is the y-intercept.

and the point (x₁, y₁).

We know that any perpendicular line to the above one will have a slope equal to:

s = -(1/a)

So the perpendicular line will be something like:

y = -(1/a)×x + c

Now we need to find the value of c, and here we need to use the point (x₁, y₁).

If the line passes through this point, this means that when x = x₁, the value of y must be y₁.

So we can just replace these in the equation:

y₁ = -(1/a)×x₁ + c

now we can solve for c:

c = y₁ + x₁/a

Then the equation of the line is:

y = -(1/a)×x + (y₁ + x₁/a)

And as you can see, here we did not use the fact that (x₁, y₁) belongs or not to the original line, so the construction is exactly the same in both cases.

If you want to learn more, you can read:

https://brainly.com/question/23905413