Perpendiculars and Distance

hi there charger people this is mrs boyd we're talking today about perpendicular lines and the distance between them so the distance or the shortest link from point to line is the segment that contains that point and is perpendicular to the line so on the diagram we have here at the top we can put line man that's not straight line at all well anyways the segment db is the distance shortest length from point to because it's perpendicular distance it crosses line ac at right angle so perpendicular lines are always going to be the shortest distance between two point in the line we want to draw the segment that represents the distance from the given point to the line well from to we want to draw perpendicular line that's straight down from to line we have to here we're going to imagine that continues on and from to will have line that looks something like that at right angle that's awful so sorry and point to we're going to imagine that goes on as well and from to lordy my gosh there we go and that's straight line that's perpendicular line draw and find the distance between each set of lines on the graph well we've got two per two lines that are parallel the shortest distance between that those two lines we're just going to count how many lines there are one two three so the distance is three the same thing with this the graph in the middle we could count the lines we want from left to right one two three four five six seven so that distance is seven and finally we just go don't even cross line we just go one down that distance is one so the distance between any two lines that are long coordinate grid lines we're just going to count the lines between them okay now we're going to work on finding the distance between pair of parallel lines but we already know that that distance has to be the perpendicular distance so the fun part is going to be finding out what that distance is so our first step is going to be to graph these to these two lines so let's do the top one first equals 2x plus 2 our y-intercept is 2 and the slope is 2. so from that that red point we're going to go up 2 and right 1. or we can go down two and right one and then we can connect the points here now want to do the same thing for the bottom equation equals 2x minus 4. well the y-intercept is negative 4 so we're going down 4. and from there the slope is two so up two right one couple times so we can connect some points here so those are our two lines and we want to find the perpendicular distance between these lines so the slope of both of these lines is two the perpendicular slope is the negative reciprocal so opposite sign we start with sine of two the new opposite sign is negative positive two now it's going to be negative the reciprocal of 2 is 1 half so the slope of our perpendicular line is going to be negative one half i'm going to choose either of these y-intercepts to be the y-intercept for the perpendicular line that we're going to draw i'm going to choose positive 2 just because it's little bit further into our graph and might look like graph little bit easier so our y-intercept is going to be 2 that means the equation of our perpendicular line is going to be equals mx plus negative 1 half plus 2 so let's graph that line the intercept is two so it's going to be right there and the slope is negative one half down one right two down one right two so that is the perpendicular line that goes between my two poorly drawn parallel lines the next part the next step set the perpendicular equation equal to the second equation and solve the system so that's exactly what we're going to do these are both equals something equals 2x minus 4 and equals negative one half plus two we set these equal to each other let's see negative one half plus two is going to equal 2x minus 4. now we solve for i'm going to add one half to each side and that gives us 2 equals 2.5 minus four usually hate decimals today i'm going to do it then we're going to add this 4 to each side we've got 6 equals 2.5 we divide both sides by 2.5 sorry i'm out of room and now need calculator we get equals 6 divided by 2.5 and that's 2.4 so now i'm actually going to move this because we're going to i'm gonna have to do some more work here because we've only got half the solution to the system sorry just checking my work so we've got is 2.4 now we want so if is 2.4 we can say equals 2 times 2.4 minus 4 like we've got right there so 2.4 times 2 is 4.8 and equals 0.8 so our coordinate point for our yeah coordinate point that we care about in this case is going to be 2.4.8 all right so next step so where this point is guess should tell you that that is the point right here where the yellow perpendicular line crosses that bottom orange line that we want to find the distance between so the two points that we're going to care about finding the distance between are going to be this two point four point eight and zero two all right so let's scoot down just bit here on the on the page we're going to find the distance between those two points well our two points that we care about are 2.4 and 0.8 and 0 2. we use the distance formula the distance is the square root of the difference in x's so 2.4 minus 0 squared plus the difference in y's 0.8 minus 2 squared well 2.4 squared is that's definitely not right it's 5.76 and that's plus 0.8 minus 2 is negative 1.2 and if we square that we get 1.44 add those together and the distance is the square root of 7.2 and that's the answer as ugly as in disgusting that is the distance between our two parallel lines this distance right here is the square root of 7.2 and that is really unfortunately as good as it gets let's do one more and see if can confuse you less we've got two new lines we've got equals negative let's draw that we can also think of this as equals negative 1 so the intercept is zero so we're going to have an intercept at the origin and then the slopes negative one so from here down one right one down one right one or up one left one whichever way floats your boat that's not the best it's not the worst second line equals negative plus 4 or equals negative 1 whichever one helps you remember what we're doing here so the intercept is 4 we go up four and same idea down one right one or up one left one all right so we connect these dots all right let's use this bottom green equation as our reference equation this time the slope here like in both equations is negative one that means the slope of our new perpendicular line that's going to allow us to find the distance between these lines is going to be the opposite sign so plus and the reciprocal which is still 1. the y-intercept of our reference line is zero so that means our equation of our perpendicular lines would be equals and that's it now we'll set the perpendicular equation equal to the second equation in our system because they both equal that'll turn into sorry my cat is really enjoying himself back there well equals negative plus 4 that's 4. so we add each side and get the x's together and that's 2x is 4 divide by 2 and we've got is 2. we want to know what is equal to so we plug 2 back in and equals 2. if use that equation right there so our point is 2 2. now we want to find the distance between those two points those two points being 0 0 this point right here and 2 2 that one right there that we want to know the distance between these two perpendicular lines or parallel lines i'm sorry so the distance equals the change in 2 minus 0 squared plus the change in simplify bit we get 2 squared is 4 plus 4 that's the square root of 8. so our distance between the two lines is square root of eight units this has really specific process hope you totally get it hope you ask any questions if you have them and hope to see you soon