Parallel, Perpendicular and Intersecting Graphs - Revision Notes

In these revision notes for Parallel, Perpendicular and Intersecting Graphs, we cover the following key points:

• What are parallel lines?
• What condition must two lines meet to be parallel?
• What happens if two lines have the same coefficient and constant?
• How do we find the equation of a line that is parallel to another known line? What other information do we need to know?
• What are perpendicular lines?
• What is the relationship between gradients in perpendicular lines?
• How do we find the equation of a line that is perpendicular to another known line?
• How do we find the intercept of two perpendicular lines?
• What are intersecting lines? When are two lines are intersecting?
• How do we find the common point of two intersecting lines?
• How do we find the distance between two parallel lines?

Parallel, Perpendicular and Intersecting Graphs Revision Notes

In geometry, parallel lines represent those lines that go in the same direction. Obviously, the lines must not go along the same coordinates, as in this case, they would overlap and we would not be able to distinguish them from each other.

The main feature of parallel lines is that they don't have any common point. In other words, parallel lines extend to infinity on both sides but they never touch each other; moreover, parallel lines always have the same distance separating them.

Two parallel lines have the same steepness, and therefore, the same gradient m (m₁ = m₂ = m). On the other hand, since the graphs do not pass through the same coordinates, they have a different y-intercept. This means the constant n of their corresponding equations is different (n₁ ≠ n₂).

Perpendicular lines form right angles around the intercept point. The relationship between the gradients m₁ and m₂ of the two given perpendicular lines is

We can also find the intercept point of the two graphs by solving the system of linear equations that represent the two lines.

We can find the following things about a line that is perpendicular to a known line:

1. The equation of the perpendicular line if it has a known point
2. The equation of the perpendicular line if the intercept point of the two lines is known
3. The intercept of the two graphs if the equations of the two perpendicular lines are both known

Intersecting lines are those lines that intercept each other at a single point but not at right angles. Therefore, in this case, we cannot find the gradient (and the equation) of the other line if one of the lines is known unless we don't have at least two known points of the second line given, where one of them may be the intercept of the two lines.

We draw four important conclusions about the relationship between linear graphs:

1. Parallel lines have the same gradient and no intercepts. Hence, the system of linear equations formed by them has no solution.
2. Perpendicular lines represent a special case of intersecting lines, where the product of the corresponding gradients is -1.
3. If two lines have different gradients m₁ and m₂, and their product is different from -1, they are simply intersecting.
4. If the coefficients m and constants n of two lines are the same, they overlap.

The distance between two parallel lines represents the shortest path that connects them, which corresponds to the distance between the two intercepts with the perpendicular line that intersects the two original parallel lines.

We can use the intercepts A and B to find the equation of the missing line if two out of three lines have known equations. Thus, if the equations of the parallel lines L1 and L2 are known, you can use any of point A or B to find the equation of the perpendicular line L3. On the other hand, if the equation of L1 and L3 are known, we need the coordinates of point B to find the equation of L2, and so on. However, the most important thing to find in this situation regards the distance between two parallel lines if both of them have known equations. The procedure used for this purpose is as follows:

Step 1: Choose an x-coordinate for any of the parallel lines and find the corresponding y-coordinate. In this way, you identify a point of that line.

Step 2: Find the gradient of the perpendicular line using the relationship between gradients in perpendicular lines.

Step 3: Find the equation of the perpendicular line given that the point identified in step 1 belongs to the perpendicular line as well.

Step 4: Make a system of equations with the equation of the perpendicular line you found in the previous step and the other parallel line, not the one of step 1.

Step 5: Solve the linear system to find the intercept of the perpendicular line to the other parallel line. This is the second point identified, after that identified in step 1.

Step 6: Calculate the distance d between the two lines by using the equation

where x₁ and y₁ are the coordinates of the leftmost point from those identified in steps 1 and 5 while x₂ and y₂ are the coordinates of the rightmost one.