What shape do you get if you joining the midpoints of the sides of the shape shown below?

What is the coordinate of the midpoint of the line segment shown below?

Forming shapes from midpoints

﻿Hi there and welcome to another lesson with me Dr. Saada. In today's lesson, we will be looking at forming shapes from diagonals. For today's lesson, you will need a pen and paper and a ruler. So grab this and let's make a start. For today's try this, I would like you to complete the code, to move the robot from one point to the other. If you look at the grid, you have few points that have been given to you along with the coordinates of the points. The robot is at point A at the moment. I want you to write the code to move the robot from A to B, B to C, C to D, D to E and E back to A. And I want you to think about describing a different loop that visits each point on that grid. So if you're feeling super confident about this, remember we've done this in lesson two and three. If you're feeling super confident about it. Pause the video and have a go. If not, don't worry I'll give you some hints. Okay, and for support, this is what you need to know first that if we are describing a movement going up, we go, we say, North, if we are going down then it is South. Well done. And if we're going to go right, it is East and the left it's West. So we have South, East and West. Okay. Now I want to move the robot from point A to point B. So I'm going to start by drawing a dotted line, okay or a line, or you can even count the lines but drawing the lines really, really help. So I'm going to draw a line from point A to B. Now to get from A to B, I need to move six units to the right. Okay. That's six units to the right. And then I need to move up. And how many units I'm at negative two, I need to get to three. That is five units up. Okay. So now when I want to write it as code, I'm going to write down is six, because we've got six units to the right, and then I'm going to write down North five. And that is the code that we'll get the robot tool from point A to point B. Now using this, can you write a code to move the robot now from point B to C. Pause the video and have a go. Now it's time for us to mark and then try this together. So we already said that for the first one our answer was from A to B is East six and then North five. What did you get for B to C? Excellent job. West five and North one. What about C to D? Great. East six and South six. And next one D to E. Good job. West three, North three and E to A. Really good. West four followed by South three. And you've managed to answer all of these questions just by drawing these straight lines, connecting the points and seeing how many steps we need to go to the right, left, up or down. Really good. Now for the second part of the challenge I wanted you to think about describing a different loop that visits each point. What did you come up with? Okay. Now this is what I have done. I thought when robot is at point A, let's say the robot was to be, next. Where can the robot go next? Can go to C, can go from C to E not necessarily D. Doesn't have to be in alphabetical order. And then from D to E and therefore the robot has visited each point. We've done a whole loop. Now, if we want the robot to go back to A, we can say D to A and the robot would go back. So we've done a different loop that's slightly differently. Excellent job. If you've done this correct, let's make a start on the connect task. Ciara is trying to prove that the shape is a rhombus. Help Ciara prove that this is a rhombus. What is a rhombus? That's the first question that you should be asking yourself. Excellent. A rhombus is a quadrilateral. I can say that a rhombus is any four sided shape with all equal sides. So the four sides in a rhombus are equal. Now, if you look at the diagram, we don't know the length of each of the sides, we have not been given the length. But what we have been given is a coordinate of each vertex. So each vertex, we know all the coordinates of it. Well it's also drawn on a grid so it's even easier for us to try and figure it out. So what we're going to do is to start with. I have already drawn some right angled triangles around to connecting each vertex. So let's have a look. We start with the first triangle. If we start with this triangle here. Okay. Let's start with this. It's a two by four. Okay. So it's two up and four across. Let's go to the second triangle. So let's look at this one here as our second triangle. It's also a two by four, and let's look at the next one. We have two by four and the last one again we have a four and a two. So to get from one vertex to the other, we always have to do a journey of four and two or two and then four. So we're really covering the same distance, and this shows us that these four sides are equal. So now I can write this down. The diagrams shows four equal right angled triangles with each side of the rhombus as the longest side of the triangle. So in each case, the longer side is the side of the rhombus. So if we have similar triangles, the triangles are exactly the same. They're identical with two and four, then the longer side must always be equal. Now I can write down the sides are all equal. So it's really important with this question, because it's a proof question. We set out the proof starting from what we need to prove, ending with the statement of what we have proved. So that's how you answer questions that say prove as the key question or key word for the question. Again. Now, carrying on with the idea of having the rhombus or using a rhombus, the line segment shows a diagonal of a rhombus. So now if we have a grid. I showed you a line segment. We know it's a line segment, it has two ends. I gave you the coordinates of each of each end. One of them has six one, and the other has a coordinate of three, four. Help Ciara find three rhombuses for the diagonal and record the coordinates of the vertices. So if I want to use this. I've got this diagonal I want to draw rhombuses around it. How can I make a start? Let's help Ciara with this. So to start with, we know that the rhombus has four equal sides. We know that the diagonals bisect each other. So they cut each other in half. So ideally I need to know the midpoint. I need to know what half of the diagonal is, because that would be the key to me answering this question. So I started thinking, okay, to get from the point three four to six one, what do I do? I need to follow this triangle that I drew here. Okay. So I need to go one, two, three down. So I need to go three down here and then one, two, three across. Okay. Now, if I want to find the midpoint, I need to do half of that. Okay. And half of that. I need to do one and a half down and then one and a half across. Okay. And this tells me that this point here is the midpoint of the line segment that I already have. So I can draw a line along there. I can draw a line there to show that, okay, now my second diagonal. So this is one diagonal, the other diagonal of the rhombus is this one here. Okay. And like it bisect each other. They cut each other at this point. Now let's get rid of what I have written already just to make it a little easier for us. Okay. Now I have the two diagonals, not just one. So all I need to do is draw the sides. Now, remember the sides are equal and we need to think about right angled triangles just like we did in the previous task. So I can start with this line for example, now I can say, well for this line, what am I doing? I'm going one. And then I'm going one, two, three, four. Okay. And this is my right angled triangle with the longer side being the first side of the rhombus. So now if I want to do the second line of the rhombus to draw it, I need to do another right angled triangle that has sides one and four. And make sure that the longer side of the triangle is actually the other side of the rhombus. So I can go here and do this. And what I've done here again, I went one, two, three, four. Okay. And then I went one up. Okay. So I did a four by one. And now again, the longer side of the triangle is the rhombus. Can you now have a little think, where would the second, the third line or the third side of the rhombus be? Really good. So I'm going to be thinking about going on the other side and doing exactly the same thing. So I'm going to go one again by four. And the longer side of the triangle is the side of the rhombus. Now I can really just connect them or I can double check that I'm doing it correctly 'cause I should have one, two, three, four, and then another one here. So now I have a rhombus. I can now write down. It said here to record the coordinate so I can record the coordinates. I already have two coordinates. Okay. I have two the coordinates of two vertices. So this point here, this Vertex here has the coordinate of what. Really good. Two zero. And this point here, what coordinate does it have? Excellent. If I read the x axis here is seven and on the y axis it's five. So the coordinate is seven five and remember the brackets. Okay. Now the question also is to find the three rhombuses. It's not just once we found one, we found the coordinates of the vertices. So let's find another one. So we're going to go back to the point three four. And from that I'm going to draw a side, a line. Now this would be the side of the rhombus, and it's really important that I think carefully about the triangles I'm creating with that because I want to copy them to find the other sides. So I just drew a line from three four. And you need to think really carefully. Okay. What have I done to get from here. From this point to where to this point, to where the new vertex is going to be. So I went down one, two, and then across one. So I need to do the same thing from this point here to get to that next vertex. So this is what I'm going to do. I'm going to draw this and that is one down and two across. Remember that it has to be the same thing, one and two or two and one to create those triangles. Now, from this point, I'm going to do the same thing or I can go back to the three four and draw a line from there, which represents the same thing so there we go. That's my next one and my last one. And now I have another rhombus. Now I want to record the coordinate of this new one. What's the coordinate of this vertex here. The coordinate is four two. So I'm going to write it down here. And the coordinate of that other vertex is going to be here. The lines would have crossed here and that would be at five three. Okay, so this and five three. So these could be two possible vertices. Okay. Now let's say we want the third one. So again, we're going to go back to that original point three four and think how can I draw a line from that? So I'm going to draw this one. Now this time, this one is interesting because I didn't get to a corner of a square. So the coordinate is going to be decimal number but that's fine. It does not have to be an integer. So again, you need to think carefully about what have I done to get from again from three four to this new vertex here. So I went down one, two, three, not quite three. I went one, two and then two and a half, and then I went across with just over a half. Okay. So I need to repeat the same thing on this side. If I do that, it gets me there. Okay. And we have created the next one. So now I know that this is my vertex. So I'm just going to drop this off. So it's easier for me to show you where the vertex is. So this is the vertex of the new rhombus. I can do exactly the same thing on the other side. So there we go. And there we go. And now this is the new vertex here. Okay. And I can write the coordinate when this one here. If we look at it, it's between three and four, it's closer to four than three. So I can say it's roughly three point, let's say 3.7. And the one coordinate is about 1.5. Sorry, 1.5. And the other vertex is again between five and six. So I would say again, maybe 5.7 and the y coordinate is 3.5. Now I am saying here it is roughly this because I don't have, it's not a draught paper. So I cannot see all the smaller lines. Okay. And it depends on where your line ends. You probably want to end it halfway in the square so you can write the coordinate as 0.5, because it's easier. Okay. I'm just going to show you one more. We've answered the question. We already have three rhombuses, but I just want to show you that we can actually draw so many. There's unlimited number of rhombuses that we can draw here, especially when we stop thinking about all the decimal coordinates that we can get. I can draw this line followed by this, followed by this and followed by that. And again, write the coordinates. And I think this one is probably closer to 0.5 or 0.45. Okay. So I just wanted to show you that there's an unlimited number of sides that I can draw to make unlimited number of rhombuses the most important thing is thinking from the first vertex. What did I do to get to the second? What was the journey along the right angled triangle. Repeat that journey. Repeat it again. And repeat it again until you have the four sides of the rhombus. It is time now for you to have a go at this independent task. You've got two questions to answer. Please pause the video and have a go at them and when you're ready to mark and correct your work, then press play again. Off you go. And for the solutions, now let's have a look at the first question. Explain why this shape is a rhombus. We've just had the discussion about the fact that rhombus, a rhombus has four equal sides. So I need to be able to prove that those four sides are equal. How do I do that? Can do it by drawing excellent right angled the triangles around them. So I have a first, like my first right angled triangle. The second one, I've drawn the third and the fourth. So I'm drawing right angled triangles to connect the vertices. These four right angled triangles are equal. They have a base of one, a height of three. And the longer side is always the side of the rhombus. So now I can say it is a four sided shape with four equal sides. Question two. This line segment is one diagonal of a rhombus. So we know we've got one diagonal of a rhombus. Give the coordinates of two points, which could be the other vertices of a rhombus. So if you have done the same steps that we have done in the connect task previously, you should have, or you could have something that looks like this. So for example you can have the vertices as zero two, four four, but there are so many possible answers. I wonder which ones you came up with. And now it's time for our explore task. And I absolutely love this question. Which of the line segments below are equal in length? I gave you six line segments and I gave you the end points of each. I gave you the coordinates of each endpoint. And I want you to use the coordinates to find out which line segments here are equal. There's more than one equal path. If you're feeling super confident about this, please pause the video and have a go at now. If not, don't worry. I'll give you some help. Okay and for help, this is what I want you to think about. I really want you to think about right angled triangles and what we have been doing throughout this lesson. So let's look at the first line segment here. And I need to draw a right angled triangle connecting the two end points of the line segment. Now I want to know how did I get what's the journey from one end point to the other? Or I can think about it as what's the height of this triangle. What I had one ordinate of four up here, and I have a one ordinate of negative one. So I went from four all the way to negative one. How many units is that? Really good. That's five units. And about going across or basically the base of the triangle if you'd like to think about it like this. What have I, where did I go from and to? I went from negative seven 'cause that first point had an x coordinate of negative seven. I went to negative five. So how many units did I go to the right? Really good. I went two units. So now I know that to get the first line segment to get from one end to the other, I need to try again. That has five and two units. Okay. So a base and a height of five and two. Now let's look at the second one. I can do exactly the same thing. Start by drawing the right angled triangle. Think about what's happening to the x moving from negative one to three. How many units is that? And think about what's happening along the y axis moving from two to five. How many units is that? Now with this hint, you should be able to complete the second line segment and do exactly the same thing for the remaining four. And then pair up the ones that are equal in length. So pause the video now and have a go at this. And now let's mark and correct our explore task. You should have now already drawn the right angled triangle connecting the two end points of each line segments and recorded down how many units you had to go across the x axis or across the y axis to get from one end to the other. Then you need to be able to compare the triangles that you have. Which two triangles are identical here? So if I look at the triangles, my first triangle it has a five and a two. A base and height of five and two. Do I have any other triangle with base and the height of five and two? No. So this one doesn't match with any. Let's look at the second one. I have a four and a three. So base and height of four and three. Do we have another one? Yes in fact we have the one on the neath it it's got four and three as the base and height. So these two, the triangles are identical. They are equal. Therefore the two line segments are equal in length. Next one, we have a four and two, okay. Base and height of four and two of the triangle. And we have the same thing here. So again, this one here, I'm going to give it two ticks matches with this one with the two ticks. So they, the two line segments here are identical. They are equal. The last one is three two doesn't match with any. So now I can write down that B equal to E and D is equal to C. And if you've done this, you should be so proud of yourself. Good job. And that is all from me today. I hope you enjoyed today's lesson. Please remember to complete the exit quiz, have a lovely day. I'll see you in the next lesson. Bye.

Which code will move the robot from A to E?

Which of these lines are the same length?

Which line segments have equal length?

What is the coordinate of the midpoints of the line segment shown below?

The line segment below is a diagonal of a rhombus. Which pair of  co-ordinates could be the vertices of the rhombus?

Forming shapes from diagonals

In this lesson, we will be comparing line segment lengths and use this to solve shape problems involving diagonals.


Maths

Y7 Maths

Key Stage 3

Year 7

Coordinates

Geometry

THEME

UNIT

The 2-D coordinate axis

Line segments

Finding midpoints

Solving geometric problems

Equations of lines

Exploring horizontal and vertical lines

What are the co-ordinates of point C?

Which point has the co-ordinates (5, -2)?

Describe the movement from B to A.

Which statement best describes line segments A and B?

Fill in the gap: The ............. describes the horizontal location of a given point.

What is the coordinate of point C in the image given below?

Which sentence correctly compares the lengths of the two line segments shown below?

Find the mid-­point of the points on the number line:

Fill in the gap: The equation .......... describes a line through every point with x-coordinate 4.

What is the equation for line C?

Lesson overview:Forming shapes from diagonals

Core Content

Coordinates: