Parallel and perpendicular lines
In our next example, we will find the equation of a line given a point on the line and the equation of a parallel line.Įxample 2: Finding the Equation of a Line given a Point on the Line and a Parallel Line Hence, since the lines have the same slopes and are not coincident, they must be parallel which is option A. They cannot be the same line, so they are not coincident. Since the two lines pass through different ?-intercepts, We can see that the first line has a ?-intercept of 9 but the second line has a
Since both lines have the same slopes, they are either Therefore, the second line also has a slope of 1 7. We can rearrange the second equation to get The first equation is already in this form, so it has slope 1 7. The form ? = ? ? + ?, where ? is the slope and ? is the ?-intercept. We can do this by writing both equations in To determine if the lines are parallel or perpendicular, we first want to find their slopes. How would you describe the relation between the lines ? = 1 7 ? + 9 and − ? + 7 ? + 4 = 0? We can do this by sketching any nonvertical lines, includingĮxample 1: Identifying Whether Two Lines Are Parallel, Perpendicular, or Otherwise
PARALLEL AND PERPENDICULAR LINES HOW TO
We can now ask the question of how to check if two lines are perpendicular. Since the two lines are distinct and have the same slopes, we can conclude that they are parallel. Would actually be the same line (called coincident lines). If they had the same ?-intercept, then the two lines The ?-intercepts of the two lines are different. The coefficient of ? is − 3, so its slope is also − 3. We can subtract 3 ? from both sides of the equation of the second
The first line is given in this form, so its slope is given by the coefficient of We can recall that a line given in the form ? = ? ? + ? has a slope of ? and a For example, consider the lines ? = − 3 ? + 2 and 3 ? + ? = 1. This allows us to check if two lines are parallel. If two distinct lines have the same slopes ( ? = ? ) or are both vertical, then they are parallel.
If two nonvertical lines are parallel, then they have the same slopes. If they are perpendicular, the product of the slopes will be −1. You can also check the two slopes to see if the lines are perpendicular by multiplying the two slopes together. If the slope of the first equation is 4, then the slope of the second equation will need to be for the lines to be perpendicular. Two non-vertical lines are perpendicular if the slope of one is the negative reciprocal of the slope of the other. The slopes of the lines are the same and they have different y-intercepts, so they are not the same line and they are parallel. Look at b, the y-value of the y-intercept, to see if the lines are perhaps exactly the same line, in which case we don’t say they are parallel. The first line has a y-intercept at (0, 5), and the second line has a y-intercept at (0, −1). The given lines are written in y = mx + b form, with m = 6 for the first line and m = 6 for the second line. Two non-vertical lines in a plane are parallel if they have both:Īny two vertical lines in a plane are parallel.ĭetermine whether the lines y = 6 x + 5 and y = 6 x – 1 are parallel.
PARALLEL AND PERPENDICULAR LINES INSTALL
Please make sure that Java 1.4.2 (or later) is installed and active in your browser ( Click here to install Java now)įrom this exploration, you may have noticed the following. Sorry, the GeoGebra Applet could not be started. For the last equation you try, can you predict what the slopes of the parallel and perpendicular lines should be? O As you try more equations, look for the relationship between the slopes of parallel lines, and the slopes for perpendicular lines. What do you notice? Look at the slopes of two perpendicular lines. O Look at the slopes of the two parallel lines. (Be sure to move your cursor slowly.) When the lines are parallel or perpendicular, text will appear to let you know you’ve done it! O Then, click and drag the dot on the red line to make the line parallel or perpendicular to the blue line. O Click and drag the dot in the “Equation” slider to choose one of five example equations. Perpendicular lines are also everywhere, not just on graph paper but also in the world around us, from the crossing pattern of roads at an intersection to the colored lines of a plaid shirt.Įxplore lines in the interactive diagram below. These 90-degree angles are also known as right angles. Perpendicular lines are two or more lines that intersect at a 90-degree angle, like the two lines drawn on this graph. Examples of parallel lines are all around us, such as the opposite sides of a rectangular picture frame and the shelves of a bookcase. Parallel lines are two or more lines in a plane that never intersect. Exploring Parallel and Perpendicular Lines