Let's solve each part of the problem one by one.

Solution:

a) The y-coordinate of any point on the x-axis:

On the x-axis, the y-coordinate of any point is always $0$.
Answer: $y = 0$.

b) The x-coordinate of any point on the y-axis:

On the y-axis, the x-coordinate of any point is always $0$.
Answer: $x = 0$.

c) The coordinates of the origin:

The origin is the point where both the x and y coordinates are $0$.
Answer: $(0, 0)$.

d) The y-coordinate of any point on the line through $(0, 1)$, parallel to the x-axis:

A line parallel to the x-axis has a constant y-coordinate for all points. Since it passes through $(0, 1)$, the y-coordinate of every point on this line is $1$.
Answer: $y = 1$.

e) The x-coordinate of any point on the line through $(1, 0)$, parallel to the y-axis:

A line parallel to the y-axis has a constant x-coordinate for all points. Since it passes through $(1, 0)$, the x-coordinate of every point on this line is $1$.
Answer: $x = 1$.

f) A line is drawn through $(1, 2)$ perpendicular to the x-axis. What are the coordinates of the point at which the line cuts the x-axis?

A line perpendicular to the x-axis is a vertical line. For a vertical line passing through $(1, 2)$, the x-coordinate remains constant at $1$ for all points on the line.

To find where this line intersects the x-axis, we set $y = 0$ (since any point on the x-axis has $y = 0$).

Thus, the coordinates of the intersection point are $(1, 0)$.
Answer: $(1, 0)$.

Would you like a detailed explanation on any part?

Related Questions:

1. What is the general equation for a line parallel to the x-axis?
2. How do you find the equation of a line given two points?
3. What is the relationship between perpendicular lines in terms of their slopes?
4. How would the answer change in (f) if the line was instead perpendicular to the y-axis?
5. What is the equation of a line that passes through the origin and has a slope of 1?

Tip:

Remember, lines parallel to the x-axis have equations of the form $y = c$ and lines parallel to the y-axis have equations of the form $x = c$.