To find the negative reciprocal, first find the reciprocal and then change the sign. As with parallel lines, we can determine whether two lines are perpendicular by comparing their slopes, assuming that the lines are neither horizontal nor perpendicular. The slope of each line below is the negative reciprocal of the other so the lines are perpendicular.
Two lines are parallel lines if they do not intersect. The slopes of the lines are the same. Coincident lines are the same line. Two lines are perpendicular lines if they intersect at right angles. Given the functions below, identify the functions whose graphs are a pair of parallel lines and a pair of perpendicular lines. Parallel lines have the same slope. Possible Answers: Yes, because the product of their slopes is not.
Correct answer: No, because the product of their slopes is not. Explanation : If the slopes of two lines can be calculated, an easy way to determine whether they are perpendicular is to multiply their slopes. Possible Answers: Yes, because the product of their slopes is. Correct answer: Yes, because the product of their slopes is.
Explanation : The product of perpendicular slopes is always. Since line passes through and , we can use the slope equation: Since the two slopes' product is , the lines are perpendicular. Are the following two lines perpendicular: and. Explanation : For two lines to be perpendicular they have to have slopes that multiply to get. Explanation : If lines are perpendicular, then their slopes will be negative reciprocals.
First, we need to find the slope of the given line. Copyright Notice. Karl Certified Tutor. College of Wooster, Bachelors, Mathematics. Malone University, Masters, Educational Technology. Ana Maria Certified Tutor. Erik Certified Tutor. Report an issue with this question If you've found an issue with this question, please let us know. Do not fill in this field. Louis, MO Or fill out the form below:.
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Also, you may want to review the information on perpendicular bisector , which won't be covered in this article. When dealing with perpendicular lines specifically, there are three general "theorems" that we can use to give us helpful information to solve more complex problems. Below are the three theorems, which we will be used later on in this article to make some proofs:. If two lines intersect to form a linear pair of "congruent angles", the lines are therefore perpendicular.
Congruent angles are just angles that are equal to each other! If two sides of two "adjacent acute angles" are perpendicular, the angles are therefore complementary. Adjacent angles are angles that are beside each other, whereas acute angles, as you hopefully recall, are angles less then 90 degrees. Now that we've defined what perpendicular lines are and what they look like, let's practice finding them in some practice problems. Looking at the lines r and p, it is clear that they intersect at a right angle.
Since this is the definition of perpendicular lines, line r is therefore perpendicular to line p. Looking at the lines r and q now, it is also apparent that they intersect at a right angle. Again, since this is the definition of perpendicular lines, line r is also perpendicular to line q. Lastly, let's take a look at the lines p and q. In the image, we can clearly see that lines p and q do not intersect, and will never intersect based on their slopes. Therefore, we can conclude that lines p and q are not perpendicular, but are instead parallel.
Solving this problem is similar to the process in Example 1. Look at the angles formed at the intersection. Since the angles are congruent, leading to perpendicular angles, according to Theorem 1 discussed earlier, the lines m and n are therefore perpendicular.
Let's take a look at lines a and b first. Clearly, as we have practiced in early examples, these two lines do not intersect, and are parallel, not perpendicular. Next, consider the lines b and c. In this case, that is equal to.
Therefore, the correct answer is:. Given the equation of a line:. When looking at the equation of the given line, we know that the slope is and the y-intercept is. Any line perpendicular to the given line will create a 90 degree angle with the given line. So in other words, the line we are looking for will have no dependence on the y-intercept, as any y-intercept will do. What we do care about is the slope of the line.
The slope of any line perpendicular to a given line has a negative reciprocal of the slope. So for this problem:. Many of the answers are reciprocals or negative slopes, but the slope we are looking for is. That leaves us with 2 answers. However, one of the answers is the exact same equation for a line as the given equation. Therefore our answer is:. Which of the following is perpendicular to. Two lines are perpendicular if and only if their slopes are negative reciprocals.
To find the slope, we must put the equation into slope-intercept form, , where equals the slope of the line. We begin by subtracting from each side, giving us. Next, we subtract 32 from each side, giving us.
Finally, we divide each side by , giving us. We can now see that the slope is. Therefore, any line perpendicular to must have a slope of. Of the equations above, only has a slope of.
Which of the following equations is perpendicular to? Convert the given equation to slope-intercept form:. Divide both sides of the equation by :. The slope of this function is :. The slope of the perpendicular line will be the negative reciprocal of the original slope. Substitute and solve:. Only has a slope of.
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