Question

Describe at least two similarities between constructing a perpendicular line through a point on a line and constructing a perpendicular through a point off a line.

Answer 1

For a given line, a perpendicular to that one intersects it forming four angles of 90°.

To answer this, to name two similarities, we will see that the construction is exactly the same in both cases, so to say (trivially) two similarities is that:

We will find the slope in the same way

We will find the y-intercept in the same way.

Now let's see how to construct perpendicular lines.

We know that a general line is given by:

y = a*x + b

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

A perpendicular line to the above one must have a slope equal to the opposite of the inverse of the above slope, thus the perpendicular line will be something like:

y = -(1/a)*x + c

Where we still need to find the value of c.

So the first similarity is that in both cases we will have the same slope (as this does not depend on the fact that this line passes through a point on the line or off the line)

Now, also remember that two perpendicular lines always do intersect, thus the perpendicular line will always pass through a point of the original line.

Now, let's assume that we know that our line passes through the point (x₁, y₁), we will see that it does not matter is this point is on the original line or not.

To find the value of c, we replace the values of the point in the line equation:

y₁ = -(1/a)*x₁ + c

c = y₁ + (1/a)*x₁

Then the line is:

y = -(1/a)*x + y₁ + (1/a)*x₁

Notice that we did not specify if (x₁, y₁) is on the line or not, thus the construction is exactly the same in both cases.

Then to, trivially, say two similarities we could say:

• We will find the slope in the same way
• We will find the y-intercept in the same way.

If you want to learn more, you can read:

brainly.com/question/10323528