What are the co-ordinates of point C?

What are the co-ordinates of point D?

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

Describe the movement from B to A.

Which statement best describes line segments A and B?

Find the coordinates of the remaining vertex for the square.

What is the midpoint of the line segment AB?

The line segment is one of the sides of a square. If you draw the full square, which of the following could could be the vertex of the square?

Solving geometric problems

﻿Hello and welcome to another math lesson with me, Dr. Saada. Today, we would be looking at forming shapes from midpoints. All you need for today's lesson is a pen and paper. So please grab these and when you're ready, let's make a start. To start today's lesson, I would like you to try this. Find the midpoints of the sight of these shapes. Join the mid points to form new shapes. We learnt how to find midpoints for line segments in lesson three. So if you're feeling confident about this, please pause the video and have a go. If not, don't worry, I'll give you some supports. And for support, you have a couple of hints here. If we look at the triangle and we look at the base. So looking at that line segment, to perform one end to the other, I need to move forward to the right. I want to find the midpoint. Instead of moving forward, I need to do half of that. So half of that journey, which is only two to the right, it tells me that the midpoint must be here. Remember the midpoint cuts that line segment into two equal parts. So now this part here And this part are equal, okay. Now second hint. Let's look at the second shape. To get from one to the other, one point to the other point. I need to go up four and to the right two. So if I want to find halfway through or the midpoint, instead of going up by four, I'm going to go up by two. Well done. And instead of going to the right by two, I'm going to go to the right by one, really good. So up two, to the right one and that is the mid point, okay. Now pause the video and have a go. Now let's go through the solutions, so you can mark and correct your work. So the first one we had the triangle. And we worked out that the midpoint of the base was here because to get from one point to the other, you move four units. So half way you move two units. Now to go up, usually you go six points. So half of that, you go three points or three units, okay. So our midpoint is here. Now I'm looking at the slanted side of the triangle, that longest side. For this one, we need to draw a triangle. If I draw a triangle, I got up and then left six by four. So if I want to do half of that, I need to go up three units and then left two. So I'm doing half of the journey and therefore, this is the midpoint. Now the question asks you to connect the midpoint and see what shapes do you form? What new shapes are we making here? So I have three points here. If I joined them together, they give me a triangle. So I can see that I have a right angled triangle. I use the midpoints, join them up together, and that gave me an enclosed triangle. Now let's look at the second shape. What's the name of this shape? Excellent. It's a rhombus, it has four equal sides, okay. And now let's have a look. We need to start with one vertex and see what is the journey to get us from one to the other. So yeah, from this first vertex to this second vertex, we go down four and to the right two. If I want to find halfway and need to go down two and one to the right, okay. So half of that journey. And therefore this is the midpoint. Next one, accidentally it's the same thing. We go to the right two and we go half forward. So now I want to do half of that journey. So I go one to the right and two up. And therefore this is the midpoint. So it always join half that triangle. Next one, we got up four and to the left two. So half of that up two and to the left one, and this is the midpoint. And the last one we will, two the left and for that we do half of that, which is one to the left and two down. And therefore, this is the midpoint. Now let's connect the mid points. What did you get? Good job. If you connect them, you get rectangle. So we have a rhombus, if find the midpoint of the rhombus and we connect them together, that gives us an enclosed rectangle. The last shape. What's the shape here? Really good, it's a rectangle. Now with this one, it's a lot easier because all of the lines are horizontal and vertical. So it's easy for us to find it where at halfway or the midpoint is. The first one, the midpoint is here. And the mid point is here. And the mid point is here. And the last one. Well done if you had this correct. Now we need to connect them. What shape do they make when you connect the midpoints? Excellent, they make a rhombus. So if I connect the midpoints of a rhombus, I get a rectangle. If I connect the midpoints of the rectangle, I got a rhombus. Excellent job, well done. And now our connect task. The shapes to the right were created by connecting the midpoints of different quadrilaterals. What could the original shapes have been? So this is really interesting if we think about the previous tasks for this one. So in the try now, I gave you the shapes and I asked you to find the midpoint and connect them. Here, I have the midpoints connected and I dropped off the original shapes. And I want you to think, what would the original shapes be here? How would they look like? So some of you probably started imagining and thinking that I probably have something that goes like this, yeah. Really good. Can you think of anything else? Any other shapes? What about with this one? So some of you are probably thinking of something like maybe this, yeah. How do you make sure we draw it and it's accurate. That's what we need to practise today. Okay, now let's have a look at this and see how are we going to draw this accurately, okay. So we're starting with the first shape. This is the shape created by joining the midpoints. So these points here, this point, this point, this point, and this point, all of these points, all the midpoints. So we went to know the original shape that we dropped off. I can start with this line and say, okay, well I have this line here and this is the midpoint. So it's right in the middle of the midpoint. I have two units to this side and I have two units to this site. Now to connect it from here to there, what do I need to do? I need one, two units so I can draw two units. Now drawing two units, got me to that midpoint. Now I need to move from the midpoint and finish that line segment. So I need another two units. Excellent. Because this midpoint here will cut this line segment to two equal parts. So I'm going to draw another two. Now I finished my line segment. I can see that the midpoint is right in the middle. I've got two here, and I've got two units here. Now from this end of the line segment to get to the midpoint, I need again two units. So I'm going to draw that. And this gets me to the middle. This is a halfway through my journey. So I need to finish that journey off by another two units by joining them on the two units. Now we do exactly the same thing. I finished that journey. This is the midpoint. So to get to the middle, to get to the midpoint, I need to go off two. So I need to do that. And to finish that journey off, I need another two. And now I'm just going to get rid of this. So you can see how the shape looks like, okay. So what do we have here? We have a square. Is it a square? How do we check? One, two, three, four units. One, two, three, four units. So we have four equal sides. So we have a square and inside it what do we have? Yeah, you're correct. We have another square inside it. Now I'm going to ask you to look at exactly the same shape. Could we have done something slightly different or a bit more creative than just a square? Okay, let's make a start with this. What about if I draw this? Now think carefully about what the midpoint is actually telling us. So this here is the midpoint of this line segments and this line going to end somewhere there, okay. So what have I done to get to this point? I went up one, two, three, and then one. So now this got me halfway through the journey, I want to finish that journey. So I need to go one, two, three and then one. So the next point is going to be here and that's where my line is going to end. So if I connect them, that's my next line. So now I have one line and this is the midpoint. This part is equal to this part, I'm happy with this. What do I need to do next? Good. I can do the same with this point. I can start from this end and say, okay, what did I do to get to this point? I went one, two, three and up one. So this caught me halfway, it got me to the mid point because this is my midpoint. I need to finish that journey. So what do I need to do? One, two, three, really good. And then I'll plan. Now I can draw my next line. Okay. And my next line should really finish somewhere here. Okay, we can see that it's outside the grid. Now what do I need to do next? Okay, let me indicate. I can decide how my shape is going to look like. And I've drawn this line here. What is this starting to look like? What kind of shape is this? What type of quadrilateral is this? Really good. So if I connect net them like this, I have a kit. Okay. Now that square that we had is not fully enclosed, so it's not completely inside the kit. It's partly enclosed and part of it is outside. But it's just a creative way of looking at which points and line segments. Now let's look at the rectangle. Okay. And now we have our rectangle. Again, these points here are all the midpoints, okay. That can be connected up and raised or dropped off the original shape. And I want us to work out together. So to start with this one, I can think of. Okay. This is the main point, I'm going to draw a line segment. What's my journey like from the beginning of the line segment to the mid point? It's one. And then one, two. So one across and then two. So to finish off that line segment, I need to do the same thing from the midpoint to the end. So that's one and then one, two. So the next point is going to be here. That's where my line segment is going to finish. And I draw it. Now, what can I do next? Okay, good. I can say, okay, how do I get to the next midpoint? So that's one, two and then one across. So two down and one across. Now to finish it off, what do I need to do? Exactly the same, finish that journey. One, two and one across. So now the next point can shoot here. Now, how do I get from here to the next midpoint? What's my journey like? I need to go one across and then two up, one, two. I've reached the midpoint so I need to do exactly the same journey, which was one and then one, two. So I finished off somewhere here. Can we start seeing what this shape is? Okay, really good. Now let's get to the midpoint again. How do I get from here to the midpoint? Or do I need to do? I need to go one, two, and then one across. two up and one across. And now I need to do exactly the same. I can already see what it's going to go, but just to double check, but I'm not making that mistake. I should be able to go one, two up and one cross, and that should get me to the end of it. Really good. What shape did we make? Really good. We've made the rhombus and there is rectangle enclosed inside it. And now let's look at the same rectangle and see if we can draw in a different shape around it, okay. So we've got the same rectangle made from midpoint. Let's see if we can make a different shape to the rhombus. And we just start with this line here and now I'm going to ask myself, the question is, how do I get from here, from the beginning of the line segment to the midpoint. I go one to the left and one down. So to end that line segment I need to, or to end that journey I need to go one to the left and then one down. Which means my line ends here. Now, how do I get from this end of the line segment to the midpoint? I go to the right one and then one down, okay. So now this is have reached half way through my journey. I need to finish it off. So I need to go again to the right one and then one down. And this gets me to this point here. Now, how do I get from this end of the line to the next midpoint? What do I do? I go down one, two, three. So point three down and one to the right. That gets me here. So I need to finish off that journey by doing exactly the same thing. I need to go one, two, three, and you can see that it's outside the grid that I have. And then one across, okay. And now I can finish drawing that. Okay. So have drawn it. Now this line here is exactly the same as this line. So we've moved one, two, three and one across. So I need to do the same thing. One, two and three and one across. And that gets me here. And now you can see that we made a different shape. This time we met at kite. Now it's okay for my shape to be outside the grid. In fact, that's why I've done it. We don't have to constraint ourselves to be inside the grid all the time, okay. We need to understand that with all the line we work on the grid and we can also work off the grid. It works, unless the question says specifically that you have to stick to the grid. Okay. And now it is your turn to have a go at the independent task. If you're feeling confident, please pause the video and have a go at questions, one and two. If not, don't worry, I'll give you some help. Again for support, let's read the question together. The midpoint of one side of a triangle is at four one and that side is shown. What are the coordinates of the vertices of the triangle? Okay. If we look here, we have already been given this as a midpoint and we have one line that has been given to us, okay. Now, think to get from one end of the line, to the other end of the line, we first need to get to the midpoint. What do we do? We went up by one and then one, two across. This gets us to the midpoint. Now we need to do exactly the same journey to get to the end of that line. So we go one up and then again, two across to the left. And that gets us to here. And this is going to be the first of those vertices. You know we're looking for for the vertices. This is our first vertex. Now have a little think. To get from this vertex, which is at, what's the coordinate here? Two two, okay. So I have two two here. How do I get from here to the next midpoint? What do I need to do? It gets me halfway through it. Doesn't get me to the end of the side. So we need to do the same thing. We need to repeat it here. So what is it that we've done here? What's the journey like from this point two two to two five? Repeat that journey to find where is the second vertex of the triangle. And then do exactly the same to see how you get to four four. For question two, a kite with vertices at zero three, six one, six five and eight three. So we need to draw that kite. You've not been given axes here, so you need to start by drawing your Y and X-axis, okay. So draw your X-axis and your Y-axis. Looking at the points that have been given to us, the four points here. What's the highest X value? Well, the highest X value X here is zero, six, six, eight. The highest X value is eight. So I would go up to at least eight or maybe even nine. Let me just make this a bit clearer. So I would go to eight or nine, probably going up in ones. And on the Y-axis what's the highest Y coordinate that we have? This Y is three, one, five and it's three here. So highest is five. So you can go on the Y-axis up to five or maybe six. And I would suggest that you go up in ones. Once you have your X and Y axes, plot those four points and connect them together to make your kite. Then the question says what shape is formed by joining the midpoints of the sides of the kite. So for each side that you draw, find the midpoint and join them together and list that coordinates of the vertices of that shape. Okay. Now, with this support, you should be able to have a go at this question. So pause the video and try questions one and two for me, please. Okay. Now it's time for you to mark and correct your work. For question one, we've been given one of the sides of the triangle. And we said, we want to know how to get from the end of that side to the midpoint, two five. Now, to get to point two five, I need to go up three units. That gets me to that midpoint. To get to the end of that side. So to get to vertex, I need to do exactly the same journey. So I need to go another three units up. And that gets me to this point here marked with the pink marker. Now I can connect that side. So I've got one side here. Now, to be honest, because it's a triangle, I can just connect now this pink point to the point six zero or I can double check that I've not made a mistake by saying, well, how do I get from this point now to four four? To do that, I need to go across to the right two. And I need to go down four. That gets me halfway through the line. So I need to finish that journey off. How do I do that? I need to go another two to the right and then another four down. Exactly the same journey. And now I can connect them. The question said, what are the coordinates of the vertices of the triangle? So this here is the first vertex and it's at, the ordinate is two two. This pink one here is the second vertex and it is at two eight. So that's the second one. And the last one is this one here, and that is at six zero. So I have six zero here. Well done if you had this correct. Now for question two, I told you that you need to draw an X-axis and Y-axis, and you need to start by plotting the kites. Once you've done that, you need to find the midpoint of each side. If you've done that correctly, you should have something that looks like this. What shape is formed by joining the midpoints? Well, if we look at the middle, we've joined the midpoint and we have a rectangle. What are the coordinates of it's vertices? What did you get? Really good job. I got three four, seven four, seven two and three two. You should be really, really proud of yourself. Well done. And here we are with our explore task. Draw the shapes described below. And I've given you four shapes to draw. The first one is a rectangle, with specific set of vertices. The second one is a triangle. The third one is a kite. And the last one is a rhombus. This is a big opportunity for you to practise coordinates and drawing the shapes and recognising the properties of these shapes. Once you're done, I would like you to think about what shapes are formed by joining the midpoint of the sides of the shapes that are listed above here for you. So once you've drawn each shape, go to each of its sides, find that where the midpoint is, connect the midpoints and see what new shapes you can form. We've done a lot of practise that is similar to this during today's lesson, but this is an opportunity for you to take it a step further. To look at the properties of the shapes that are enclosed. What is happening? Can you make any connections? Are we always getting the same enclosed shape or other different? Start having a little, think about those questions. Off you go. Okay, and if you manage to draw the shapes accurately, you should have something that looks like this. And if you figured out where the midpoints are and joined them, for the kite, you should end up with rectangle enclosed, for the rectangle, you should end up with a rhombus enclosed. For the triangle, you end up with another triangle enclosed. And for the rhombus, you end up with the rectangle enclosed. A huge well done if you managed to draw these shapes accurately and managed to find out that enclosed shapes in each of them. The shapes enclosed by joining the midpoint of those sides. Its really, really important that our questions similar to this on your exit quiz. And this brings us to the end of our maths lesson for today. I would like you to take couple of minutes to reflect back on today's lesson and decide what are the two most important things that you've learned from today's lesson. Once you're done with that, I would like you to complete the exit quiz. And that is it for me for today. Have a lovely day and see you in the next lesson, bye.

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

In this lesson, we will be finding and using midpoints to solve various shape problems.

Maths

Y7 Maths

Key Stage 3

Year 7

Coordinates

Geometry

THEME

UNIT

The 2-D coordinate axis

Line segments

Finding midpoints

Forming shapes from diagonals

Equations of lines

Exploring horizontal and vertical lines

Which line segments have equal length?

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 midpoints

Core Content

Coordinates: