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I can calculate the length of a line segment on a coordinate grid in all four quadrants or using the coordinate pairs.

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Licence

This content is © Oak National Academy Limited (2023), licensed on Open Government Licence version 3.0 except where otherwise stated. See Oak's terms & conditions (Collection 2).

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Key learning points

Any line segment can be turned into a right-angled triangle by adding two lines which meet at 90°
Calculating the horizontal distance gives the length of one side
Calculating the vertical distance gives the length of the other side
These two shorter sides lengths can be used to calculate the length of the line segment
This can be done using Pythagoras' theorem

Common misconception

Pupils may think that direction matters when determining the distance between two points.

Direction matters when dealing with displacement, not distance.

Keywords

Hypotenuse - The hypotenuse is the side of a right-angled triangle which is opposite the right angle.
Pythagoras' theorem - Pythagoras’ theorem states that the sum of the squares of the two shorter sides of a right-angled triangle is equal to the square of the hypotenuse.

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Hello, I'm Mrs. Lashley and I'm gonna be working with you as we go through this lesson today.

I hope you're ready to try your best and give it all a go as best as you can, and I'm gonna help you along the way.

So our lesson outcome today is to be able to calculate the length of the line segment on a coordinate grid in all four quadrants or using the coordinate pairs.

So we're gonna be looking at line segments on coordinate grids or without the grid.

So we're gonna be using some keywords during the lesson.

You've met them before, but just in case we need a bit of reminder, I'm gonna read them out to you and explain them a little bit more.

So the hypotenuse is the side of a right angle triangle, which is opposite the right angle.

Because the right angle is the largest angle in the right angle triangle, then this is also the longest edge.

Pythagoras's theorem states that the sum of the squares of the two shorter sides of the right angle triangle is equal to the square of the hypotonus.

So the two shorter edges and the hypotenuse, it's the hypotenuse because it's a right angled triangle.

So today's lesson on calculating the length of the line segment is going to be broken into two learning cycles.

That first learning cycle is looking at calculating the length of line segments on a grid.

So thinking about a coordinate grid and those four quadrants.

And the second one is looking at line segments using only coordinate pairs, so not having the diagram to support us.

So let's make a start at looking at that first one where we try to calculate the length of line segments using coordinate grids.

So on the screen, we can see a grid background and we can think of those as one unit squares and we've got two line segments.

We've got line segment AB, and we've got line segment CD.

So which line segment is the longest out of those two? Pause the video and think about that.

Think about what strategy you can be sure to know which one is longer.

Press play when you're ready to move forward.

The line segment CD is the longest.

How do we know that? Well, if we look at their horizontal distances, AB is a horizontal line segment and we can use the gridded background to count how long it is.

And it is eight squares long or eight units.

If we look at CD, it's also got a horizontal distance of eight units, but it's also going up.

So it's got to be longer in order to cover the horizontal distance of eight units, but also be higher up.

So we can use that grid background there to support us to know that that would be the longer line segment.

Okay, so what about these two? So we've got AB and CD as line segments once again.

So which line segment is longest now? So pause the video, think about potentially horizontal distances, but maybe vertical distances too.

Press play when you're ready to check.

So neither of them are longer than the other.

They have the same length, so they look different.

They've got different gradients if we think about the steepness and the direction of them, but they have the same length.

And why is that? Well, once again, if we think about horizontally, horizontally, they are both eight squares long, but they also have this vertical dimension that is three squares.

So they're both eight squares horizontally and three squares vertically.

So now, we've got three line segments to decide between.

We've got line segment AB.

We've got line segment CD.

And we've got line segment EF.

So pause the video, look at those, think about strategies you might have in order to be able to determine which one is the longest.

Press play when you're ready to move on.

So AB is eight units.

We can see it's eight units from just a horizontal line segment using the squares.

CD is not a horizontal line segment nor is it a vertical line segment, which would be as easy as AB to calculate.

We can think about the horizontal and vertical elements.

And here, we can see a right angle triangle.

And the similar idea for E and F.

That's not the only place that you could have drawn those horizontal and vertical marks.

Can you think about where else it would be and why they would give you the same answer? So they could be on the other side of the diagonal line segment and they would give you the same answer because you'd be constructing or you'd be making a rectangle, and a rectangle can be split into two congruent right angle triangles.

So let's look at that CD a little bit closer.

So if we've put that right angle triangle where we've thought about the horizontal line segment and the vertical line segment, this vertical part is six units and the horizontal part is two units.

That's not the length of the line segment.

But we know that if we think of it like a translation to get from that first point, we need to go two units to the right and six units up.

But the direct distance from that point to the other point, we need to calculate, and we can do this using pyrosis theorem because horizontal and vertical are perpendicular directions and therefore, there would be a right angle.

So two squared plus six squared, that's the sum of the two shorter squares.

And we know that that is equal to the square of the hypotenuse.

So if we square root it, that will give us our hypotenuse or our length of the line segment.

That's 6.

32 units when square rooted and rounded to two decimal places.

So that line segment CD, we have calculated the length.

So the AB was eight units that was longer than 6.

32.

So here is a check for you.

Here is EF.

So work out the length of the line segment EF.

Pause the video and when you're ready to check your answer, press play.

So if we, first of all, we need to think about those horizontal and vertical distances.

And then we can use Pythagoras' theorem because they are perpendicular to each other and EF would be our hypotenuse.

So six squared plus five squared, you may have done five squared plus six squared and then square root it to get the length of EF, and that's 7.

81 units to two decimal places.

So if we go back to the question, which one is the longest? Well, AB actually was the longest.

We couldn't compare them in the same way that we had previously because they had different horizontal lengths and different vertical lengths.

So instead, we did need to know the actual length of the line segment.

So this first task.

On question one, I'd like you to find the length of each line segment that you can see on that grid to two decimal places.

So making use of the fact that horizontal and vertical are perpendicular, that we're on a square background, our grid is squared and therefore perpendicular distances.

And using Pythagoras' theorem.

Pause the video and then when you're ready to move on with this task, press play.

So here is question two.

And you need to find the length of each line segment and give your answer to two decimal places.

The line segments are now on the diagrams. You can see GH, JK, and LM.

Pause the video and then when you finished with those three parts and press play.

We'll go through our answers to task A and see how you've got on.

So question one, you needed to work out the length of each line segment to two decimal places.

AB would be 5.

83 to two decimal places.

And you can see that that triangle that we should have made use of was a triangle with a length of three and a length of five.

So A would be three and B would be five, or you may have had them the other way round.

CD.

So CD is the hypotenuse of a triangle that has shorter edges of seven units and three units.

So using Pythagoras' theorem, the length of CD is 7.

62 to two decimal places.

And finally, EF is a hypotenuse of a triangle with an edge of 10 and an edge of 2.

So using Pythagoras' theorem, we can see that the length is 10.

20 to two decimal places.

It is important that you've put that zero because if it states two decimal places, then your answer should have two digits after the decimal point.

On question two, you needed to find the length again of the line segments.

So for A, it was GH, and you can see the triangles on this one.

So it was 6.

71 to two decimal places.

For B, JK was 4.

12 to two decimal places.

And finally, C, LM was 5.

83 to two decimal places.

Once again, you didn't need to draw your triangle in the same orientation.

You may have gone up from the G and then right to the H, but because of the congruent triangles within the rectangle, then you will get the same answer.

That GH is the diagonal.

So we're now up to the second learning cycle and this time, we're gonna remove that grid.

We're gonna remove the diagram and we're gonna try and find the length of line segments still, but using only the coordinate pairs, the coordinates of the end of the line segments.

So here, we have two points, A and B, and a speedboat, and someone says, I'm stuck in flatland, can you help me? How far away from point B in miles am I? I'm currently at 3,7 and my destination is at 8 to 15.

So let's think about those coordinates.

So they're stuck in flatland, we're in a two dimensional space, but they do know the coordinate that they've started or they were currently at was 3,7 and they know where they destination is 8, 15.

So we need to think about how that is a change, how has that coordinate, 3,7 and then to get to 8, 15.

And we're gonna think about using a little bit like a translation.

How has that mapped from that start point to that destination point? Well, on the x-coordinate.

So that's how horizontal coordinate, if we remember, it has changed by five.

And vertically, on the y-coordinate is changed by eight.

They're both positively change, they're both an increase.

So if we do think about a diagram of what this is suggesting, well, the start point A, where that person is currently situated on their speedboat and where the destination point, they are going to need to go across five units.

And we can see that from our change in the x-coordinate and they need to increase by eight on the y-coordinate.

So the question was how many miles away from their destination from point B are they? Well, we can see that our X and our y-coordinates are perpendicular.

The x-axis and the y-axis are perpendicular.

So it is a right angle triangle, which means that we can make use of Pythagoras' theorem.

So this is our hypotenuse, this line segment between A and B.

There's this distance from where they are to where they're trying to get to.

So we can use Pythagoras' theorem, a five squared plus eight squared and square rooted to get the length of the line segment, which is the hypotenuse, which is 9.

43.

So the distance, Sam says, in response is 9.

43 miles.

They said thank you.

So they now know how far away from their destination they are.

That's obviously gonna be helpful if they're running out of fuel and they need to worry if they're gonna get there or not.

So let's have a look at one that's out of context just two coordinate pairs.

So A is the coordinate 6, 20 and B is the coordinate 4, 7.

So if we look at those two and think about that as our start and our end of our line segment, how do we change to get from one to the other? Well, our x-coordinate has gone down by two and our y-coordinate has also gone down and this time by 13.

So we're gonna use Pythagoras' theorem to work out the length of that line segment.

You might be concerned about the fact that they are negative, but remember when you swear a negative, you get a positive.

So Pythagoras theorem will still hold.

And if you were to sketch this out as a diagram, you still will find yourself a right angle triangle.

It will just be the orientation of the triangle that this negative is going to have an effect on.

But the actual length horizontal change is a positive two in terms of a distance.

So if we've used Pythagoras theorem, it's 13.

15.

So here's one for you to do.

So what is the length of the line segment that connects coordinates A and B? Pause the video and when you're ready to check, press play.

So again, think about how they've changed.

The x-coordinate has increased by two and the y-coordinate has increased by six.

So this would be our horizontal distance and our vertical distance.

Remember they're perpendicular, so we've got this right angle and we can think of AB the line segment as our hypotenuse.

So using Pythagoras' theorem, hopefully, you have calculated the AB is 6.

32 units to two decimal places.

So onto the last task of the lesson, you've got two questions.

They're both on the screen here.

So on question one, you need to find the length of each line segment.

Give your answers to do to decimal places.

So you've got six parts there, A through to F.

If you have a look, F is including some algebra, but think about it carefully and you'll be absolutely fine.

If you need to draw or sketch yourself a diagram, then, of course, you can.

But just thinking about the change and what that might means in terms of your horizontal and vertical should be enough.

And then question two, this one needs you to have a little bit of a think and maybe a diagram might be a good way to go into question two.

Find the length of line segment CD starting at the origin.

So think about what that word means mathematically.

With midpoint 9, 1.

Give your answer to two decimal places.

So pause the video, work through question one and question two.

And then when you're ready to check your answers, press play.

So we've got all of the answers to question one on the screen here.

So AB is 4.

24 and that's come from finding the difference between the x-coordinates.

So there's 3 to 6 as a difference of three, and the y-coordinate is also 3 to 6.

So that's a difference of three.

So there, your two shorter edges of the triangle that you'd be creating, thinking about horizontal distance and vertical distance.

It's an isosceles in fact.

And then we would use Pythagoras' theorem.

So three squared plus three squared gives you the square of the hypotenuse.

And so we square root that.

And that's where the 4.

24 comes from.

Then B is also 4.

24.

If you have a look, the two coordinates are the other way round.

So the difference is -3 and -3, but the distance that they are, the horizontal distance is three units.

And so we get the same calculation to work out AB.

The answer to C is 12.

73, the answer to D is 6.

71, and the answer to E is 6.

08, all given to the two decimal places.

And then if we look at F, the answer is five.

So I said that there was some algebra, the x-coordinate was P on the one end of the line segment and P plus four on the other.

So what's the change? Well, the change is four.

On the y-coordinate, it's changed from six to nine, so that's a change of three.

So it was still the same process despite the fact that there was now some algebraic expressions involved in your coordinates.

So hopefully, you were successful on question one.

And then we're gonna look at question two, which is slightly different.

So this time, you needed to find the length of the line segment CD.

So that's what we've been doing this whole lesson, but the information given was slightly different.

So it says starting at the origin.

So we needed to remember that the origin is the coordinate 0, 0, and it says the midpoint is 9, 1.

Well, what does midpoint mean? Well, this is the point that is exactly halfway along the line segment.

It is the same distance from C as it is to D.

So if we say that C is the origin, 0, 0, then D is this point two times as far as the midpoint is from C.

So this midpoint is nine to the right and one up from the origin because that's what the coordinates 9, 1 means.

So if we go another nine to the right and one up, we would find ourselves at the coordinates 18, 2, and that will be the point D.

So now, we've got our star coordinate of the line segment, which is 0, 0, and we've got the finish of the line segment, which is 18, 2.

So this time, we can now use Pythagoras' theorem.

And so you can use 18 because that is the difference between the two x-coordinates and square it 'cause that's your horizontal change.

And two is the difference between your y-coordinates, so square it 'cause that will be your vertical change.

And sum them together because Pythagoras' theorem is the sum of the two squares of the shorter sides and then square root it to get the hypotenuse or in this case, the length of the line segment CD.

It's 18.

11 to two decimal places.

An alternative method that you may have done is you may have met, you may have worked out the distance from the origin to the midpoint and then doubled it because the distance from one end to the midpoint is the same from the midpoint to the other end.

The thing you need to be more mindful of this method is the rounding error will increase.

So if you rounded your distance from C to the midpoint and then you double it, there's gonna be further rounding error.

So if you've got an answer very close to 18.

11 and you use the other method, that might be the reason.

So to summarise today's lesson on calculating the length of a line segment.

So any line segment can be turned into a right angle triangle by adding two lines which meet at 90 degrees.

Calculating the horizontal distance will give you the length of one side.

Calculating the vertical distance gives you the length of the other side.

These two shorter side lengths can then be used to find the length of the line segment because that will be the hypotenuse and this will be done using Pythagoras' theorem.

So really well done today and I look forward to working with you again in the future.

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, for this right-angled triangle.", "image": "https://www.thenational.academy/_next/image?url=https%3A%2F%2Foaknationalacademy-res.cloudinary.com%2Fimage%2Fupload%2Fv1722686198%2Fo9qc5yj207imhccn5icr.png&w=640&q=75", "imageKey": "b0e60307-92e0-d7c3-df75-3828569e3869/question/e091be2682e36dd5f29ad70da00d68f7.png" }, "questionType": "Short answer", "correct": [ { "text": "42 cm" }, { "text": "forty two" }, { "text": "42" } ], "id": "7bddbe6d-364d-611b-959c-e04588443d56", "lessonId": "9e860666-1448-788e-2d64-f85ca546e86f" }, { "type": "Question", "question": { "text": "M is the midpoint of a line segment AB. Given that A(4, 8) and B(6, 20), what are the coordinates of the midpoint M?" }, "questionType": "Multiple choice", "correct": [ { "text": "M(5, 14)" } ], "choices": [ { "text": "M(4, 14)" }, { "text": "M(5, 10)" }, { "text": "M(5, 14)" }, { "text": "M(6, 12)" }, { "text": "M(1, 6)" } ], "id": "c00b0eda-c082-18ad-03e8-99131efdb416", "lessonId": "9e860666-1448-788e-2d64-f85ca546e86f" }, { "type": "Question", "question": { "text": "M is the midpoint of a line segment AB. Given that A(5, 9) and M(8, 5), what are the coordinates of B?" }, "questionType": "Multiple choice", "correct": [ { "text": "B(11, 1)" } ], "choices": [ { "text": "B(6.5, 7)" }, { "text": "B(3, -4)" }, { "text": "B(2, 13)" }, { "text": "B(11, 1)" }, { "text": "B(1, 11)" } ], "id": "a100a665-0fba-a67e-15b9-ce0a2e3e5d18", "lessonId": "9e860666-1448-788e-2d64-f85ca546e86f" } ], "lessonId": "9e860666-1448-788e-2d64-f85ca546e86f", "meta": { "gradeStageId": "ks4", "subject": "Maths", "tier": "Foundation", "grade": "Year 10", "topic": null, "unit": "Right-angled trigonometry", "lessonCount": "11", "age": [ 14, 15 ], "gradeId": "age_14_15", "gradeStage": "GCSE", "lesson": "Calculating the length of a line segment" } }, { "id": "0cdbdb04-fef7-b763-6a9d-b98c5f698b66", "type": "Resource", "resourceType": "Quiz", "title": "Calculating the length of a line segment", "quizType": "Exit quiz", "items": [ { "type": "Question", "question": { "text": "Put the length of these line segments in order, starting with the shortest.", "image": "https://www.thenational.academy/_next/image?url=https%3A%2F%2Foaknationalacademy-res.cloudinary.com%2Fimage%2Fupload%2Fv1722686197%2Fbeigvis5eb69xtv0ar7h.png&w=640&q=75", "imageKey": "0cdbdb04-fef7-b763-6a9d-b98c5f698b66/question/e770c3f44e8b9269e00ebd2f4cfb1ea3.png" }, "questionType": "Ordering", "elements": [ { "order": { "text": "1" }, "item": { "text": "c" } }, { "order": { "text": "2" }, "item": { "text": "a" } }, { "order": { "text": "3" }, "item": { "text": "b" } } ], "id": "49b278e1-87bb-4fce-ecd4-67f30381d314", "lessonId": "9e860666-1448-788e-2d64-f85ca546e86f" }, { "type": "Question", "question": { "text": "What is the length of the line segment AB, to 1 decimal place?", "image": "https://www.thenational.academy/_next/image?url=https%3A%2F%2Foaknationalacademy-res.cloudinary.com%2Fimage%2Fupload%2Fv1722686197%2Fzftzaiu9z60jphc47yka.png&w=640&q=75", "imageKey": "0cdbdb04-fef7-b763-6a9d-b98c5f698b66/question/33cfa393668250a2c8383d3b6d3d85c0.png" }, "questionType": "Short answer", "correct": [ { "text": "7.6 units" }, { "text": "7.6" } ], "id": "fbc6a5e1-3493-02e9-0ee1-2a9909db8ad8", "lessonId": "9e860666-1448-788e-2d64-f85ca546e86f" }, { "type": "Question", "question": { "text": "What is the length of the line segment AB, to 1 decimal place?", "image": "https://www.thenational.academy/_next/image?url=https%3A%2F%2Foaknationalacademy-res.cloudinary.com%2Fimage%2Fupload%2Fv1722686197%2Fl8kciildgzsqrgdppbrm.png&w=640&q=75", "imageKey": "0cdbdb04-fef7-b763-6a9d-b98c5f698b66/question/f09dead8afcbf34fbb07b548a1c0fca2.png" }, "questionType": "Short answer", "correct": [ { "text": "5.8 units" }, { "text": "5.8" } ], "id": "bf7ec7ca-7b22-fab4-1e85-1714215b17e7", "lessonId": "9e860666-1448-788e-2d64-f85ca546e86f" }, { "type": "Question", "question": { "text": "Given that A(4, 7) and B(7, 11), what is the length of the line segment AB?" }, "questionType": "Short answer", "correct": [ { "text": "5 units" }, { "text": "5" }, { "text": "five" } ], "id": "bc43b893-a05b-33db-0bbc-a54ebb826d47", "lessonId": "9e860666-1448-788e-2d64-f85ca546e86f" }, { "type": "Question", "question": { "text": "Given that A(-9, -9) and B(3, -4), what is the length of the line segment AB?" }, "questionType": "Short answer", "correct": [ { "text": "13 units" }, { "text": "13" }, { "text": "thirteen" } ], "id": "c97fac08-cec5-0981-39f2-087d63661642", "lessonId": "9e860666-1448-788e-2d64-f85ca546e86f" }, { "type": "Question", "question": { "text": "Find the length of line segment CD starting at the origin with midpoint (4, 2). Give your answer to 2 decimal places." }, "questionType": "Short answer", "correct": [ { "text": "8.94 units" }, { "text": "8.94" } ], "id": "335db26e-9c8c-b3b2-5bf3-3ad44116e2f0", "lessonId": "9e860666-1448-788e-2d64-f85ca546e86f" } ], "lessonId": "9e860666-1448-788e-2d64-f85ca546e86f", "meta": { "gradeStageId": "ks4", "subject": "Maths", "tier": "Foundation", "grade": "Year 10", "topic": null, "unit": "Right-angled trigonometry", "lessonCount": "11", "age": [ 14, 15 ], "gradeId": "age_14_15", "gradeStage": "GCSE", "lesson": "Calculating the length of a line segment" } } ] }