Perpendicular lines meet at a____________.

What is the midpoint between the coordinates (3,5) and (9,11)?

What is the midpoint between the coordinates (3,5) and (7,9)

Which of these coordinates is the same distance from (3,5) as (7,9)

Perpendicular bisectors

Key Stage 3

Maths

Constructions, congruence, and loci

﻿Hello, and welcome to another video. In this lesson, we'll be talking about perpendicular bisectors. My name is Mr. Maseko. Before you start this lesson, make sure you have a pen, a pencil, a ruler, and something to write on. You will also need some loose sheets of paper. So if you don't have those things, go get them now. Okay. Now that you have these things, let's get on with today's lesson. First, try this activity. So you're going to have to follow the instructions really, really carefully. And this all starts with you drawing a triangle and labelling the vertices A, B, C. And then make sure you follow those instructions very, very carefully. Pause the video here and give that a go. Okay. Now that you've tried this, let's see what you should have come up with. Well, after you follow the instructions, so you folded so that A meets B. And then you marked where the fold line meets AB, and then you fold it so that A meets C and you've marked where the fold line meets AC, and same thing for B and C. You should have ended up with a picture that looks something like this. So this is the point with a fold line that you had met AB. And then this is the point where the fold line met BC. So that's when you joined B and C. And then this is the point where your fold line met AC. So that's after you joined A and C. Now, you could have then ... Let's label this. So this is ... We'll label this as AB, and then this as BC dash. Let's use the dash so that we don't confuse that. And then this is AC dash. And then if you did it again, so made AB meet BC, and then you put a dot in the middle, etc, etc. You'd have ended up with something that looks like this. Now, what do you notice about these points? What do you notice about this point and the line AB? But what you should have noticed is that that point is the midpoint of the line AB. And then that point there is the midpoint of the line AC. And then that point is the midpoint of the line BC. Now, if you repeated these steps, something really special happens. So if you notice this point here, that's in a line with that point, so that fold line also goes through that second point that you would have made if you repeated these steps. Same thing with what? This point here. And then the same thing with this point there. Now, there is something really special about those lines that I've just drawn, and the lines BC, AB, and AC. What do you think that is? Well, turns out that those lines meet these lines at a 90 degree angle. So those lines are perpendicular. And they're perpendicular and they go through the midpoint of those lines. So those red lines that I just drew are perpendicular, and they go through the midpoints of the lines of your triangles. So, perpendicular lines are cut the line in a half. Those lines that we've just drawn are perpendicular bisectors. So, here's what we have. So, this is the definition we're just talking about. A perpendicular bisector is a locus of points that are equidistance from two ends of a line segment. So if we join these two points together, what we make is a locus of points that are equidistant from two ends of a line segment. And that locus of points meets your line segment at a right angle. So, as we were saying before, if we have two points that are equidistant between two ends of a line segment, we can draw a perpendicular bisector. So like, if you join up these two points, those two points make a perpendicular bisector to the line AB. And we call it a bisector because what? It cuts. Yeah. So we call it a bisector because it cuts the line segment in half. So, as we were saying, in order to construct this perpendicular bisector, we need to have two points that are equidistant from the two ends of our line segments. Now, the first point is the easiest one to find, because it will be on the line. So the first thing we want to state is the coordinate of the point on a AB that is equidistant from both ends of the line. So this coordinate we're talking about is the midpoint. Well, we have coordinates. So, if we look at this, that first coordinate is negative two, four, and then this coordinate here is negative four, negative four. And we want the midpoint of this line. And just by looking on a coordinate grid is quite easy to know what the midpoint is, 'cause you can see that the middle of this line is here at negative three, zero. Well, how do I know? Because if you look at the distance from A is four down, and one across. And then from B, one across and four up. So, a coordinate that you could find here is these coordinates, one, negative one. Now, how do I know that's the same distance from A and B? Well, 'cause if you look, to get from B to one, negative one. You go five across and three up. You see, you go five across and three up. And then to go from A to one, negative one, what do you do? You go three across and five down. So now we have two points that are the same distance from ... Let me get rid of some of these so we can see them clearly. So we said the two points were, this one and this one. So now we have two points that are the same distance from A as they are from B. So now we can do what? We can join them together and we've made our perpendicular bisector. So for this independent task, what I want you to do is to draw the perpendicular bisectors for these three line segments. Remember the instructions we had before. First, find a point on the line, so the midpoint. And another point that's the same distance from the two ends of the line segments. And then join them together, and you will have your perpendicular bisector. Pause the video here and give that a go. Okay, now that you've tried this, let's see what you could have come up with. Well, we'll start with AB. The midpoint is there, which was the point six, five. Then another point that's the same distance from A as it is from B. Well, that would have been ... You could've picked. Let's say that coordinator there and that coordinate is four, seven. How do I know? Because it's what? Four across from A and it's four up from B. And if you join those two together, you've made your perpendicular line segment. Now, pay attention to the gradients of those two lines. A gradient of those line segment AB or the gradient of that when you go up one across on the X-ordinate you go one up on the Y-ordinate. So AB has a gradient of one. So you have the gradient. And the gradient of the perpendicular bisector, while you go one across on the X-ordinate, you go one down on the Y. So the perpendicular ... I'm going to do this little right angle. Well, that gradient was negative one. Hmm. Now let's do the other one. Let's see if this pattern carries on. Well, if we look at the line segment CD. Well, the midpoint was here at zero, zero, and you could have picked a point. Let's say we pick this point. We'll pick the point three, three. Now that point three, three is the same distance from C as it is from B. And if you join those together, there's your perpendicular bisector. Now again, the gradient of CD, well, that is negative one. The gradient of the perpendicular line is one. Okay. Well, let's go on to the line segment EF. Well, this is a horizontal line. So we know that the perpendicular line will be vertical. So this is an easier one to do. The midpoint is there and anywhere on that vertical, you can make your perpendicular bisector. Now this relationship between the gradients of your perpendicular lines is really important. And I want you to start spotting patterns between the gradients of the line segments and their perpendicular bisectors. For this task, you're going to use that relationship to find the equations of the perpendicular bisectors for each of these line segments. Now, if you want a clue for how to do this, keep watching the video, otherwise pause the video in three, two, one. Okay. For those of you that want to clue. Well, the first thing is we first have to draw the perpendicular line. So, same as we did before, we want two points that are equidistance from both ends of the line segment. So let's do the line segment CD. Well, we know the midpoint of CD is here. Now we want another point that's equidistant. Well, another point that would be equidistant from the line CD, would be this point here. The point two, six. How do I know? Because it's two up and one across. Oh sorry. And four across from D. And it is four up and two across. And if you can see, those two triangles are the same triangles they've just been rotated. So, that point is the same distance from C as it is from D. And if you join that point to there, what do you have? There's your perpendicular bisector. Now, what is the gradient of that perpendicular bisector? Well, you go one across on there, you got what? Three up on the Y. So you go one across on the X, you go three up on the Y. So that gradient is three. So we know that line will be, y is equal to 3x, because the gradient is three. So it'd be y equals 3x, but is it y equals to 3x exactly? Well, what coordinates do we have with the coordinates one, three, and the coordinates two, six? And we can see that that's going to go through the coordinate zero, zero. So yes, that equation of that perpendicular bisector will be y is equal to 3x. Now do the same thing for the other lines that you can see. Pause the video now. Okay. Now that you've all tried this, let's see what it is you've done. Well, we've done the equation of that first perpendicular line. The next easiest equation to do is that blue line, because it's a vertical line. We know that the equation of the perpendicular line would just be the horizontal while the midpoint is here on the X-axis. So at any point on that X-axis will be a perpendicular line. So the perpendicular line is basically on the X-axis. That's the perpendicular bisector. I know that X-axis, we know that the equation is y is equal to zero. 'Cause anywhere on the X-axis, y is equal to zero. Now, what about ... Let's do the orange line. Well, for that orange line, what do we have? We have a midpoint there, and we have another point at say zero, zero. That point at zero, zero is equidistant from F as it is from E, because if you look from E, one across, three up, and then from F, three across, one down. Those two triangles, same triangle, just rotated. So those distances that I'm highlighting are the same. So, if you join those two points together, there's your perpendicular bisector. And then now the equation of that perpendicular bisector, well, what's the gradient. Well, the gradient is, you go one across on the x, so you go half down on the y. So the gradient is negative a half. So that would be, y is equal to negative half of x, and it goes through zero, zero. So we know that just y equals negative half of x. Coincidentally, that gradient of that perpendicular line was negative a half, the gradient of the orange line, that gradient you go one across on the x, you go two up on the y, that gradient of the orange line was equal to two. When the gradient is two, the perpendicular line or the gradient of negative a half. For that first purple line we did, the gradient to the perpendicular line was three, the gradient of the purple line. Well, that was negative one over three. Are you seeing a relationship between the gradient of the perpendicular line and ... Are you seeing a relationship between the gradients of perpendicular lines? Okay. So now let's do the green line. On the green line, well, our midpoint is there. And you want a point that is equidistant from both ends. So another point you could have had. Well, if I go five across and one down that's the point zero, zero. So again, at the point zero, zero, there's a perpendicular bisector. So what's the equation of that perpendicular line. Well, that could be, y is equal to negative one fifth of x. Well, how do I know? 'Cause the gradient, when you go five across you go one down, so it's negative one over five. A gradient of the green line that was equal to five. So look at these gradients. Look at the gradients of the perpendicular lines. Can you see the relationship? When the gradient of one line is three, the perpendicular line is negative one over three. When the gradient of one line is two, the perpendicular line is negative one over two. When the gradient of one line is five, the gradient of the perpendicular line is negative one over five. What can you say? What conjunction can you make? Well, we can see that the perpendicular gradient always has a denominator that's the same as the gradient of the other line and it's always negative one over. Can you see? Now, that's a relationship that you will see very often as you get more experience with the equations of lines. Now, thank you very much for participating in today's lesson. And again, if you want to share your work, ask your parent or carer to share work on Twitter tagging @OakNational and #LearnwithOak. Thank you for participating in today's lesson. I will see you again next time. Bye for now.

Perpendicular bisectors

Perpendicular bisectors

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UnitMaths / Constructions, congruence, and loci

Maths