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Section 5.7: Parallel and Perpendicular Lines

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alright class today we're going to talk about parallel and perpendicular lines parallel lines and what they look like you're some graph paper in here are two parallel lines and just going to say that there parallel you will notice ladies and gentleman that these two lines have a certain slope about them line has a slope of 0 and the red line has a slope of 0 I want you to take that information and be able to apply it algebraically because my recognize slopes of parallel lines you want to know that parallel lines have the same slope so if you're asked a question which of these lines are parallel  you can notice that this  line and this line are parallel because if you look at their slopes which is in slope intercept form 4/3 here 4:30 these lines have the same slope so they are parallel you don't need a graph for anything just need to look at the equation of line in slope intercept form to realize that they are parallel because they have the same slope in the same likewise matter if you look at these two lines these lines are also parallel because they're slope in this case is zero easier to both of these lines are horizontal lights okay I want to write an equation of a parallel line we want to write an equation of a line that passes through the .45 and is parallel to y equals 5x + 10 what you want to have ladies and gentleman is your reference point always want to make sure that you have your equation in slope intercept form y equals MX plus b and the first thing replace right away is M45 and why did we do that ladies in German Wells because if I want to create a parallel line that is parallel to this they need to have the same slope so since this line is parallel the slopes have to be the second and then after you do we cannot replace this .45 into your X and Y because that will satisfy this equation and give me a very specific line that is parallel to y what's 5x + 10 with a very specific Y intercept which will be subbing for so here is the equation that we just created with a slope of 5 and instead of Exile put four instead of why I'll put 5 and then you just solve for this ladies and gentlemen you going to isolate B you have 5 equals 20 plus B20 on both sides and your V is -15 now this is in your answer what you want to do is replace the be here well minus 15 and this is the equation of the very specific line that is both parallel to this one and passes through this point alright okay go ahead and positive here and practice this one on your own know when you're doing this problem you should have noticed that this equation is not in slope intercept form so that means you can't easily see the slope that you need to have you need to write your equations in slope intercept form therefore the Y needs to be isolated isolate the Y by subtracting X on both sides I'll have y equals negative X plus one and so now I can use my reference points the slope here many people would have thought the slope is one but a slope is actually negative one and once again I do that because I want two parallel lines if this line has parallel to this line that means they have to have the same slope which is -1 and then you replace your ex new y  with the play that they give you which is 2 and 3 in stead of exit foot to stead of why I put three and now I just saw for be  nde pause  replace the plus be here with + 5 and you have the equation of the line that is once again parallel to this one add it passes through this point alright okay and where they cross the creates a right angle  if you look at these slopes the slope to the red line is once again zero slope of the line is what we call undefined and we can write undefined as any number written over zero and you'll see it at one of the conditions for line perpendicular slopes needs to be reciprocals of each other because a slope of 0 can be written as a zero over any number and a slope that is Define undefined can be written as any number zero the other condition though when we recognize and slopes of perpendicular lines is that they must also be opposites some looking for opposite reciprocal slopes that is perpendicular to y Plus 8 you're a list of lines you want to be looking at slopes ladies and gentlemen this is written in slope intercept form to slope here is at 4 and I'm trying to find the opposite reciprocal of negative for the opposite of negative 4 is positive for negative positive and the reciprocal of positive for is 1/4 and so I'm looking at my options and I'm looking for an equation where the slope one fourth and there it is ladies and gentleman is in point slope form the slope is next to the quantity of x + x or 10 - x 1  and so since this line has a slope that is an opposite reciprocal of the original equation slope when is perpendicular slopes no that's practice the art of finding opposite reciprocal the number is 2/3 the opposite reciprocal is -3/2 the numbers -6/5 the opposite reciprocal of it is positive 5 6 and the numbers 10 the opposite reciprocal is -1/10 alright so you take the opposite of the positive and negative and just go ahead and take the reciprocal of the number and you have the opposite reciprocal of that number Wright's Okay so  algebraically we need to be able to write an equation for perpendicular lines we are going to write an equation of a passes through .514 and is perpendicular to y equals one-third x + 3  once again use your reference point Y equals MX you can Replace Em right away with negative 3 because if these light too much are to be perpendicular the slopes have to be opposites as well as reciprocals so -3 1/3 are opposite reciprocals and then all you have to do is replace the X and the Y with the that they want you want the light to pass through so actually be five wise 14 and you solve for your Y intercept of this new line that you're trying to create this is 15 about size and you have vehicles 29 you plug in your + 29 into the slope intercept form and you have your final equation alright ladies and gentleman soap before you do any kind of writing before you write any equation you need to make sure that the slope you have here is correct here is not negative five even though it looks deceiving because it's right next to the extern you need to remember that in slope intercept form the Y needs to be stuff is not myself here you need to divide all sides by by 10 and when you do that you have y equals negative one half X + 2  and then now you can write a perpendicular line because you know now this slope is negative one the new slope for the perpendicular line is positive too because once again you want these two lines to be perpendicular slopes need to be opposite reciprocals your place the X and the Y with 8 and 2 and you will solve for your y-intercept  and vehicles -14 you're going to plug in -14 back into slope intercept form and you have your alright okay if you have any questions please do not hesitate to come see me before after school during a lunch or during your break I'll be happy

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Section 5.7: Parallel and Perpendicular Lines

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