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.
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.
Since the theorem consists of a biconditional statement, the proof will consists in two parts.
Both directions of the biconditional statement have been proved.
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.
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
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.
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
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.
System I: Infinitely many solutions
System II: No 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
From the previous example, the following conclusions about systems of linear equations can be made.
|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.
If and are two perpendicular lines with slopes and respectively, the following relation holds true.
Since the theorem is a biconditional statement, it will be proven in two parts.
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.
The biconditional statement has been proven.
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
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
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.
Yes, see solution.
Yes, see solution.
Start by drawing the diagonals of the rhombus. Then, find the slopes of the diagonals.