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Connecting Algebra and Geometry Through Coordinates

Slope of Parallel and Perpendicular Lines

In this lesson, the relationship between the slopes of parallel and perpendicular lines will be explored.

Catch-Up and Review

Here are a few recommended readings before getting started with this lesson.

Explore

Investigating the Slopes of Parallel Lines

The lines and are positioned as shown in the graph. Move the point vertically.

Compare the slope triangles. What conclusion can be made about the triangles and the lines?

Discussion

Slopes of Parallel Lines Theorem

In the previous graph, the slope triangles and can be mapped onto each other by a translation. Since translations are rigid motions, it can be concluded that and are congruent triangles. Because the slope triangles are congruent, the slopes of the lines are equal. This means that the lines are parallel.

In a coordinate plane, two distinct non-vertical lines are parallel if and only if their slopes are equal.

Parallel Lines

Let and be parallel lines, and and be their respective slopes. Then, the following statement is true.

The slope of a vertical line is not defined. Therefore, this theorem only applies to non-vertical lines. Any two distinct vertical lines are parallel.

Proof

Since the theorem consists of a biconditional statement, the proof will consists in two parts.

  1. If two distinct non-vertical lines are parallel, then their slopes are equal.
  2. If the slopes of two distinct non-vertical lines are equal, then the lines are parallel.

Part 1

Consider two distinct non-vertical parallel lines in a coordinate plane. Their equations can be written in slope-intercept form. Suppose that the slopes of the lines are different. This means that and are not equal. Using the Substitution Method, the system of equations formed by the equations above can be solved.
Solve for
Since the expression is not undefined because its denominator is different than zero. To find the coordinate of the solution, can be substituted for in Equation (II).
The solution to the system formed by the equations was found. This implies that the lines and intersect. However, this contradicts the fact that the lines are parallel. Therefore, the assumption that the slopes are different is false. Consequently, the slopes of the lines are equal.

Part 2

Now, consider two distinct non-vertical lines and that have the same slope Their equations can be written in the slope-intercept form. Since these are distinct lines, and are not equal. With this information in mind, suppose that the lines intersect. Solving the system of equations will give the point of intersection. Again, the Substitution Method will be used.
The obtained result contradicts the fact that and are different. Therefore, there is no point of intersection between the lines and This means that they are parallel lines.

Both directions of the biconditional statement have been proved.

Pop Quiz

Practice Finding the Slope of Parallel lines

If the equation of a linear function is written in slope-intercept form, its slope can be identified.

By rewriting the given equation in slope-intercept form, find the slope of a parallel line to the line whose equation is shown. If necessary, round your answer to decimal places.

A linear equation

Example

Identifying the Equation of a Parallel Line

Consider the equation of a line written in slope-intercept form. Which of the following is the equation of the line that passes through the point and is parallel to the given line.

Hint

Start by identifying the slope of the given line. Then, use the Slopes of Parallel Lines Theorem.

Solution

Consider both the general form of the slope-intercept form of a line and the given line. In the slope-intercept form, represents the slope and the intercept. Since the given line is in this form, the slope of the line is By the Slopes of Parallel Lines Theorem, a parallel line to this line also has a slope of It is given that this parallel line passes through the point By substituting this point into the equation of the parallel line, its intercept can be found.
Solve for
The equation of the parallel line to through the point is


Example

Finding a Parallel Line When Given a Graph and a Point

Up to now, it has been discussed how to find the equation of a parallel line to a line whose equation is given. What about finding the equation of a parallel line to a line whose graph is given?

The first transatlantic telegraph cable was laid between Valentia in western Ireland and Trinity Bay Newfoundland in the With the invention of fiber optic cables, the number of transatlantic cables has increased. The following map shows some of these cables.

World Map
Kevin wants to write the equation of a line parallel to the first transatlantic cable and passes through point a specific location in Virginia beach. Kevin draws a coordinate plane, then line on which the cable lies, and the point as shown.
A Line and a Point
a If Kevin wants to use transformations to find the equation of the parallel line, what would be his next step? Use transformations to write the equation of the parallel line. Write the equation in slope-intercept form.
b Use the Slopes of Parallel Lines Theorem to find the equation of the line. Write the equation in slope-intercept form.

Answer

a Example Next Step: The next step would be determining the number of units needed to vertically translate the given line so that it passes through the given point.
b Equation:

Hint

a By which transformation can a line be mapped onto a parallel line?
b What is the slope of the given line?

Solution

a Parallel lines can be mapped onto each other by a translation. Therefore, the next step will be to determine the number of units needed to translate the line so that it passes through
A Line and a Point
As can be seen in the graph, if the given line is translated units down, a parallel line through is obtained. As a result, the values of the parallel line will be less than the corresponding values of the given line. With this knowledge, the equation of the parallel line can be found by following two steps.
  1. Find the equation of the given line.
  2. Translate the given line units down.

Step

By observing the graph, it can be seen that the line passes through the points and When a slope triangle is constructed between these points, the slope of the line is calculated as

A Line and a Point

It can be seen that the line intercepts the axis at Therefore, the intercept is Knowing the slope and the intercept of line is enough to write its equation in slope-intercept form.

Step

Recall that the values of the parallel line are less than the corresponding values of the given line. Therefore, to obtain the equation of the desired line, units must be subtracted from the obtained equation.

The equation of the parallel line to through is
b Using the points and the slope of the line is calculated as
A Line and a Point
Therefore, by the Slopes of Parallel Lines Theorem, all parallel lines to have a slope of Then, the equation of the parallel line passing through can be written as follows, where is the intercept. Since should lie on the line, the value of can be found by substituting its coordinates into the above equation.
Solve for
The equation of the parallel line to that passes through is The equation is the same as the equation obtained in Part A, as expected.

Example

Systems of Equations With Parallel or Overlapping Lines

Recall that a system of linear equations can have infinitely many solutions, one solution or no solution. Under which conditions does a system of linear equations have infinitely many solutions or no solution?

Consider the following systems of equations. Determine the number of solutions to each system without finding the actual solutions, if exist.

Answer

System I: Infinitely many solutions
System II: No solution

Hint

Determine whether the equations in the system represent parallel lines. What does this say about the number of solutions?

Solution

Start by examining the first system of equations. Dividing both sides of the second equation by will result in the first equation. This result means that the equations in this system represent the same line. Therefore, the lines are coincidental, and they have infinitely many points of intersection. Consequently, System (I) has infinitely many solutions. The equations in the System (II), however, differ in their intercepts. Since the equations are written in slope-intercept form, their slopes can be identified as

Equation Slope Intercept

By the Slopes of Parallel Lines Theorem, these lines are parallel and do not intersect. Therefore, System (II) has no solution.

Discussion

Properties of Different Systems of Equations

From the previous example, the following conclusions about systems of linear equations can be made.

Condition Conclusion Example
The lines of the system have the same slope and the same intercept. The lines are coincidental. This means that there are infinitely many points of intersection. Therefore, the system has infinitely many solutions.
The lines of the system have the same slope but different intercept. The lines are parallel. This means that there is not a point of intersection. Therefore, the system has no solution.
The lines of the system have different slopes. The lines are neither parallel nor coincidental. This means that there is one point of intersection. Therefore, the system has one solution.

These conclusions can be seen in the following diagram.

intersection of two lines in the plane

Explore

Investigating Lines Using Slope Triangles

In the graph below, the slope triangles seem to be congruent. The congruence of these triangles can be shown by a rotation. Move the slider to rotate line counterclockwise around point
Think about these questions.
  • Is it possible to find a relationship between the slopes of the lines before rotating
  • How does the rotation change the slope of line

Discussion

Properties of Perpendicular Lines

In the previous graph, it can be seen that the initial angle between the lines measures Therefore, the lines are perpendicular.

Rule

Slopes of Perpendicular Lines Theorem

In a coordinate plane, two non-vertical lines are perpendicular if and only if their slopes are negative reciprocals.

Perpendicular Lines

If and are two perpendicular lines with slopes and respectively, the following relation holds true.

This theorem does not apply to vertical lines because their slope is undefined. However, vertical lines are always perpendicular to horizontal lines.


Proof

Since the theorem is a biconditional statement, it will be proven in two parts.

  1. If two non-vertical lines are perpendicular, then the product of their slopes is
  2. If the product of the slopes of two non-vertical lines is then the lines are perpendicular.

Part

Let and be two perpendicular lines. Therefore, they intersect at one point. For simplicity, the lines will be translated so that the point of intersection is the origin.
Let and be the slopes of the lines and respectively. Next, consider the vertical line This line intersects both and
Since and are assumed to be perpendicular, is a right triangle. Using the Distance Formula, the lengths of the sides of this triangle can be found.
Side Points Length
Since is a right triangle, by the Pythagorean Theorem, the following equation must be true. The next step is to substitute the lengths shown in the table.
Simplify
It has been proven that if two lines are perpendicular, then the product of their slopes is

Part

Here it is assumed that the slopes of two lines and are opposite reciprocals. Consider the steps taken in Part This time, it should be shown that is a right triangle.

If the lengths of the sides of satisfy the Pythagorean Theorem, then the triangle is a right triangle. The side lengths, which were previously solved for in Part 1, can be substituted in the above equation.
Simplify
Since a true statement was obtained, is a right triangle. Therefore, and are perpendicular lines. This completes the second part.

The biconditional statement has been proven.

Pop Quiz

Practice Finding the Slope of Perpendicular Lines

If the equation of a linear function is written in slope-intercept form, its slope can be easily identified.

Find the slope of the line perpendicular to the given equation's line. This can be done by rewriting the given equation in slope-intercept form. If necessary, round the answer to two decimal places.

A linear equation

Example

Identifying the Equation of a Perpendicular Line

Just like with parallel lines, there is an infinite number of perpendicular lines to a given line. Two pieces of information will help in writing the equation of a perpendicular line. These are the slope — which is calculated using the slope of the given line — and a point that lies on the line.

The equation of a line is given in standard form. Determine the equation of a perpendicular line to the given line that passes through the point

Hint

Start by identifying the slope of the given line. Then, use the Slopes of Perpendicular Lines Theorem.

Solution

For any equation written in slope-intercept form the value of is its slope. Since the given equation is not written in this form, it will be rewritten so that the slope can be identified. The given equation's slope is Now, by the Slopes of Perpendicular Lines Theorem, the product of the slope of the given line and the slope of a line perpendicular must be By substituting for in the above equation, the slope of a perpendicular line can be found.
Solve for
This means that any perpendicular line to the given line will have a slope of Therefore, a general equation in slope-intercept form for all the lines perpendicular to the given line can be written. It is given that this perpendicular line passes through the point By substituting this point into the above equation, the value of will be found.
Solve for
Therefore, the equation of the perpendicular line to the line with equation through is

Example

Finding a Perpendicular Line When Given a Graph and a Point

It has been discussed how to find the equation of a line that is perpendicular to a line whose equation is given. What about finding the equation of a line perpendicular to a line whose graph is given?

Mark and Paulina have been asked to write an equation of a line perpendicular to the line shown in the graph. Additionally, they are told that this perpendicular line should pass through the point
A line and a point
Mark says that perpendicular lines have the same slope as the given line. Paulina, on the other hand, says that the slopes of a line and a perpendicular line are opposite reciprocals.
a Determine who is correct.
b Which of the following is the equation of the perpendicular line through

Hint

a What does the Slopes of Perpendicular Lines Theorem state?
b Find the slope of the given line.

Solution

a Recall that the Slopes of Perpendicular Lines Theorem states that two nonvertical lines are perpendicular if and only if they have opposite reciprocal slopes. From here, it can be concluded that Paulina is correct and Mark is not.
b To find the equation of a perpendicular line, start by determining the slope of the given line.
A line and a point
As it can be seen in the graph, the slope of the given line is By the above theorem, the product of the slope of the given line and the slope of a perpendicular line must be By substituting for into the above equation, the value of can be found.
Solve for
The slope of all perpendicular lines to the given line is Therefore, a general equation in slope-intercept form for all lines perpendicular to the given line can be written as follows. Finally, the value of the intercept needs to be found. It is known that the perpendicular line passes through the point Since the coordinate of this point is then the value of is equal to the coordinate of the point, which is With this information, the desired line can be written.

Closure

Using Properties of Parallel Lines To Classify Parallelograms

The theorems seen in this lesson can be used to identify quadrilaterals and some of their properties.

Determine whether quadrilateral is a parallelogram. Explain the reasoning.

Answer

Yes, see solution.

Hint

Recall that a parallelogram is a quadrilateral with two pairs of parallel sides.

Solution

From the graph, it appears that and are parallel and that and are parallel. To prove this claim, start by finding the slope of each side.
As it can be seen, the slopes of the sides and are the same, as well as the slopes of and Therefore, by the Slopes of Parallel Lines Theorem, and are parallel and and are parallel. Since the given quadrilateral has two pairs of parallel sides, it is a parallelogram.
Determine whether the diagonals of rhombus are perpendicular. Explain the reasoning.

Answer

Yes, see solution.

Hint

Start by drawing the diagonals of the rhombus. Then, find the slopes of the diagonals.

Solution

Start by drawing the diagonals of the rhombus. Then, find the slopes of the diagonals.
The slope of is and the slope of is Notice that their product is By the Slopes of Perpendicular Lines Theorem, it can be concluded that the diagonals are perpendicular to each other.
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