GCSE Maths: Midpoints and lengths of line segments (Foundation)
Part of the GCSE Maths straight-line graphs and coordinates chapter (Foundation). Chapter overview · Higher version of this lesson · whole chapter on one page · Download worksheet (PDF) · Answers (PDF)
Midpoints and lengths of line segments
A line segment is the part of a straight line between two points. Two questions come up all the time: where is its middle? and how long is it?
The midpoint
The midpoint is the point exactly halfway between the two ends. To find it, work on the x-coordinates and the y-coordinates separately. The number halfway between two numbers is their mean: add them and divide by 2. For example, halfway between 2 and 8 is (2 + 8) ÷ 2 = 5, because 5 is 3 above 2 and 3 below 8. So for ends (x1, y1) and (x2, y2):
midpoint = ( (x1 + x2) ÷ 2 , (y1 + y2) ÷ 2 )
The length of a line segment
To find the length of a sloping line, make it the longest side of a right-angled triangle. Draw a horizontal line across and a vertical line up (or down) between the two points. The lengths of these two short sides come from subtracting the coordinates. Then use Pythagoras’ theorem: in a right-angled triangle, the square of the longest side (the hypotenuse) equals the sum of the squares of the other two sides, c² = a² + b². You will meet this theorem in full in Chapter 20; here we use it for a single job.
Some triangles have whole-number sides, such as 3, 4, 5 and 5, 12, 13 and 6, 8, 10, and the length is an exact whole number. Otherwise the answer is a decimal on your calculator (round it as asked), or on the Higher tier you may be asked to leave it as a surd, such as √13 or 2√13.
Key ideas
- Midpoint of (x1, y1) and (x2, y2) = ((x1 + x2) ÷ 2, (y1 + y2) ÷ 2): add then halve.
- Length: find the horizontal distance and the vertical distance (subtract), then Pythagoras: length² = (across)² + (up)².
- Square first, add, then square root last: √(a² + b²) is not a + b.
- Higher: leave exact answers as surds (for example 2√13); to find an end point, repeat the step from the known end to the midpoint.
Key words. The midpoint of a line segment is the point exactly halfway along it. The mean of two numbers is their sum divided by 2.
Method. The midpoint is exactly halfway between two points. Find the mean of the x-coordinates and the mean of the y-coordinates.
- Write down the two points and pick out their coordinates. The midpoint uses the x-coordinates together and the y-coordinates together.A = (2, 3): x1 = 2, y1 = 3
B = (8, 7): x2 = 8, y2 = 7 - Find the x-coordinate of the midpoint. The number halfway between two numbers is their mean: add them and divide by 2.(2 + 8) ÷ 2 = 10 ÷ 2 = 5
- Find the y-coordinate of the midpoint. Do exactly the same with the y-coordinates.(3 + 7) ÷ 2 = 10 ÷ 2 = 5
- Write the answer as a coordinate pair. Across first, then up or down.Midpoint = (5, 5)
- Sketch to see that it makes sense. The midpoint M must be on the line, exactly halfway between the two ends.
Check. From A to the midpoint the x-coordinate changes by 3 and from the midpoint to B it changes by 3; the y-coordinate changes by 2 and 2. The two steps are equal, so M is halfway. ✓
Watch out. Add the coordinates and halve. Subtracting gives the length of the gap, not the halfway point. Work with x and y separately.
Key words. The midpoint of a line segment is the point exactly halfway along it. The mean of two numbers is their sum divided by 2.
Method. The midpoint is exactly halfway between two points. Find the mean of the x-coordinates and the mean of the y-coordinates.
- Write down the two points and pick out their coordinates. The midpoint uses the x-coordinates together and the y-coordinates together.A = (−4, 1): x1 = −4, y1 = 1
B = (3, 6): x2 = 3, y2 = 6 - Find the x-coordinate of the midpoint. The number halfway between two numbers is their mean: add them and divide by 2.(−4 + 3) ÷ 2 = −1 ÷ 2 = −0.5
- Find the y-coordinate of the midpoint. Do exactly the same with the y-coordinates.(1 + 6) ÷ 2 = 7 ÷ 2 = 3.5
- Write the answer as a coordinate pair. Across first, then up or down.Midpoint = (−0.5, 3.5)
- Sketch to see that it makes sense. The midpoint M must be on the line, exactly halfway between the two ends.
Check. From A to the midpoint the x-coordinate changes by 3.5 and from the midpoint to B it changes by 3.5; the y-coordinate changes by 2.5 and 2.5. The two steps are equal, so M is halfway. ✓
Watch out. Add the coordinates and halve. Subtracting gives the length of the gap, not the halfway point. Work with x and y separately.
Key words. A line segment is the part of a line between two points. Pythagoras’ theorem says that in a right-angled triangle the square of the hypotenuse equals the sum of the squares of the other two sides (see Chapter 20).
Method. Draw a right-angled triangle using the horizontal and vertical distances between the points, then use Pythagoras’ theorem to find the length of the line (the hypotenuse).
- Find the horizontal distance between the points (the difference in x). Subtract the x-coordinates and ignore the sign: a distance cannot be negative.7 − 1 = 6
- Find the vertical distance (the difference in y). Do the same with the y-coordinates.10 − 2 = 8
- Sketch a right-angled triangle on the line. The line AB is the hypotenuse (the longest side, opposite the right angle). The two shorter sides are the horizontal and vertical distances.
- Use Pythagoras’ theorem: c² = a² + b². Square the two shorter sides and add them to get the hypotenuse squared.AB² = 6² + 8² = 36 + 64 = 100
- Take the square root. The square root undoes the squaring, giving the length itself.AB = √100 = 10
Check. 10² = 100 = 100. ✓
Watch out. Add the squares before square-rooting. √(9 + 16) is 5, not 3 + 4.
Key words. A line segment is the part of a line between two points. Pythagoras’ theorem says that in a right-angled triangle the square of the hypotenuse equals the sum of the squares of the other two sides (see Chapter 20).
Method. Draw a right-angled triangle using the horizontal and vertical distances between the points, then use Pythagoras’ theorem to find the length of the line (the hypotenuse).
- Find the horizontal distance between the points (the difference in x). Subtract the x-coordinates and ignore the sign: a distance cannot be negative.2 − (−3) = 5
- Find the vertical distance (the difference in y). Do the same with the y-coordinates.4 − (−1) = 5
- Sketch a right-angled triangle on the line. The line AB is the hypotenuse (the longest side, opposite the right angle). The two shorter sides are the horizontal and vertical distances.
- Use Pythagoras’ theorem: c² = a² + b². Square the two shorter sides and add them to get the hypotenuse squared.AB² = 5² + 5² = 25 + 25 = 50
- Take the square root and round. The square root undoes the squaring. Use a calculator, then round as the question asks.AB = √50 = 7.0711...
= 7.1 to 1 decimal place
Check. The line should be longer than the longer leg (5) and shorter than the two legs added (10). 7.1 is in between. ✓
Watch out. Add the squares before square-rooting. √(9 + 16) is 5, not 3 + 4.
Watch out
Add for the midpoint, subtract for the distance. A common error is to subtract the coordinates when asked for the midpoint. Another is to write √(9 + 16) as 3 + 4: you must add the squares before taking the square root, giving √25 = 5.Exercise 12B Midpoints
Non-calculatorKey words. The midpoint of a line segment is the point exactly halfway along it. The mean of two numbers is their sum divided by 2.
Method. The midpoint is exactly halfway between two points. Find the mean of the x-coordinates and the mean of the y-coordinates.
- Write down the two points and pick out their coordinates. The midpoint uses the x-coordinates together and the y-coordinates together.A = (2, 4): x1 = 2, y1 = 4
B = (8, 10): x2 = 8, y2 = 10 - Find the x-coordinate of the midpoint. The number halfway between two numbers is their mean: add them and divide by 2.(2 + 8) ÷ 2 = 10 ÷ 2 = 5
- Find the y-coordinate of the midpoint. Do exactly the same with the y-coordinates.(4 + 10) ÷ 2 = 14 ÷ 2 = 7
- Write the answer as a coordinate pair. Across first, then up or down.Midpoint = (5, 7)
Check. From A to the midpoint the x-coordinate changes by 3 and from the midpoint to B it changes by 3; the y-coordinate changes by 3 and 3. The two steps are equal, so M is halfway. ✓
Watch out. Add the coordinates and halve. Subtracting gives the length of the gap, not the halfway point. Work with x and y separately.
Key words. The midpoint of a line segment is the point exactly halfway along it. The mean of two numbers is their sum divided by 2.
Method. The midpoint is exactly halfway between two points. Find the mean of the x-coordinates and the mean of the y-coordinates.
- Write down the two points and pick out their coordinates. The midpoint uses the x-coordinates together and the y-coordinates together.A = (0, 6): x1 = 0, y1 = 6
B = (10, 2): x2 = 10, y2 = 2 - Find the x-coordinate of the midpoint. The number halfway between two numbers is their mean: add them and divide by 2.(0 + 10) ÷ 2 = 10 ÷ 2 = 5
- Find the y-coordinate of the midpoint. Do exactly the same with the y-coordinates.(6 + 2) ÷ 2 = 8 ÷ 2 = 4
- Write the answer as a coordinate pair. Across first, then up or down.Midpoint = (5, 4)
Check. From A to the midpoint the x-coordinate changes by 5 and from the midpoint to B it changes by 5; the y-coordinate changes by −2 and −2. The two steps are equal, so M is halfway. ✓
Watch out. Add the coordinates and halve. Subtracting gives the length of the gap, not the halfway point. Work with x and y separately.
Key words. The midpoint of a line segment is the point exactly halfway along it. The mean of two numbers is their sum divided by 2.
Method. The midpoint is exactly halfway between two points. Find the mean of the x-coordinates and the mean of the y-coordinates.
- Write down the two points and pick out their coordinates. The midpoint uses the x-coordinates together and the y-coordinates together.A = (1, 3): x1 = 1, y1 = 3
B = (9, 7): x2 = 9, y2 = 7 - Find the x-coordinate of the midpoint. The number halfway between two numbers is their mean: add them and divide by 2.(1 + 9) ÷ 2 = 10 ÷ 2 = 5
- Find the y-coordinate of the midpoint. Do exactly the same with the y-coordinates.(3 + 7) ÷ 2 = 10 ÷ 2 = 5
- Write the answer as a coordinate pair. Across first, then up or down.Midpoint = (5, 5)
Check. From A to the midpoint the x-coordinate changes by 4 and from the midpoint to B it changes by 4; the y-coordinate changes by 2 and 2. The two steps are equal, so M is halfway. ✓
Watch out. Add the coordinates and halve. Subtracting gives the length of the gap, not the halfway point. Work with x and y separately.
Key words. The midpoint of a line segment is the point exactly halfway along it. The mean of two numbers is their sum divided by 2.
Method. The midpoint is exactly halfway between two points. Find the mean of the x-coordinates and the mean of the y-coordinates.
- Write down the two points and pick out their coordinates. The midpoint uses the x-coordinates together and the y-coordinates together.A = (4, 2): x1 = 4, y1 = 2
B = (12, 8): x2 = 12, y2 = 8 - Find the x-coordinate of the midpoint. The number halfway between two numbers is their mean: add them and divide by 2.(4 + 12) ÷ 2 = 16 ÷ 2 = 8
- Find the y-coordinate of the midpoint. Do exactly the same with the y-coordinates.(2 + 8) ÷ 2 = 10 ÷ 2 = 5
- Write the answer as a coordinate pair. Across first, then up or down.Midpoint = (8, 5)
Check. From A to the midpoint the x-coordinate changes by 4 and from the midpoint to B it changes by 4; the y-coordinate changes by 3 and 3. The two steps are equal, so M is halfway. ✓
Watch out. Add the coordinates and halve. Subtracting gives the length of the gap, not the halfway point. Work with x and y separately.
Method. The number halfway between two numbers is their mean: add them and divide by 2.
- Add the two numbers. Use brackets for negative numbers.4 + 10 = 14
- Divide by 2. Halving the total gives the number the same distance from both.14 ÷ 2 = 7
Check. From 4 to 7 is 3, and from 7 to 10 is 3. They are equal. ✓
Watch out. Halving is dividing by 2, not subtracting 2.
Method. The number halfway between two numbers is their mean: add them and divide by 2.
- Add the two numbers. Use brackets for negative numbers.(−2) + 6 = 4
- Divide by 2. Halving the total gives the number the same distance from both.4 ÷ 2 = 2
Check. From −2 to 2 is 4, and from 2 to 6 is 4. They are equal. ✓
Watch out. Halving is dividing by 2, not subtracting 2.
Method. The number halfway between two numbers is their mean: add them and divide by 2.
- Add the two numbers. Use brackets for negative numbers.(−9) + (−3) = −12
- Divide by 2. Halving the total gives the number the same distance from both.−12 ÷ 2 = −6
Check. From −9 to −6 is 3, and from −6 to −3 is 3. They are equal. ✓
Watch out. Halving is dividing by 2, not subtracting 2.
Method. The number halfway between two numbers is their mean: add them and divide by 2.
- Add the two numbers. Use brackets for negative numbers.1 + 4 = 5
- Divide by 2. Halving the total gives the number the same distance from both.5 ÷ 2 = 2.5
Check. From 1 to 2.5 is 1.5, and from 2.5 to 4 is 1.5. They are equal. ✓
Watch out. Halving is dividing by 2, not subtracting 2.
Key words. The midpoint of a line segment is the point exactly halfway along it. The mean of two numbers is their sum divided by 2.
Method. The midpoint is exactly halfway between two points. Find the mean of the x-coordinates and the mean of the y-coordinates.
- Write down the two points and pick out their coordinates. The midpoint uses the x-coordinates together and the y-coordinates together.A = (−4, 2): x1 = −4, y1 = 2
B = (6, 8): x2 = 6, y2 = 8 - Find the x-coordinate of the midpoint. The number halfway between two numbers is their mean: add them and divide by 2.(−4 + 6) ÷ 2 = 2 ÷ 2 = 1
- Find the y-coordinate of the midpoint. Do exactly the same with the y-coordinates.(2 + 8) ÷ 2 = 10 ÷ 2 = 5
- Write the answer as a coordinate pair. Across first, then up or down.Midpoint = (1, 5)
Check. From A to the midpoint the x-coordinate changes by 5 and from the midpoint to B it changes by 5; the y-coordinate changes by 3 and 3. The two steps are equal, so M is halfway. ✓
Watch out. Add the coordinates and halve. Subtracting gives the length of the gap, not the halfway point. Work with x and y separately.
Key words. The midpoint of a line segment is the point exactly halfway along it. The mean of two numbers is their sum divided by 2.
Method. The midpoint is exactly halfway between two points. Find the mean of the x-coordinates and the mean of the y-coordinates.
- Write down the two points and pick out their coordinates. The midpoint uses the x-coordinates together and the y-coordinates together.A = (−7, −3): x1 = −7, y1 = −3
B = (1, 5): x2 = 1, y2 = 5 - Find the x-coordinate of the midpoint. The number halfway between two numbers is their mean: add them and divide by 2.(−7 + 1) ÷ 2 = −6 ÷ 2 = −3
- Find the y-coordinate of the midpoint. Do exactly the same with the y-coordinates.(−3 + 5) ÷ 2 = 2 ÷ 2 = 1
- Write the answer as a coordinate pair. Across first, then up or down.Midpoint = (−3, 1)
Check. From A to the midpoint the x-coordinate changes by 4 and from the midpoint to B it changes by 4; the y-coordinate changes by 4 and 4. The two steps are equal, so M is halfway. ✓
Watch out. Add the coordinates and halve. Subtracting gives the length of the gap, not the halfway point. Work with x and y separately.
Key words. The midpoint of a line segment is the point exactly halfway along it. The mean of two numbers is their sum divided by 2.
Method. The midpoint is exactly halfway between two points. Find the mean of the x-coordinates and the mean of the y-coordinates.
- Write down the two points and pick out their coordinates. The midpoint uses the x-coordinates together and the y-coordinates together.A = (−2, 5): x1 = −2, y1 = 5
B = (4, −1): x2 = 4, y2 = −1 - Find the x-coordinate of the midpoint. The number halfway between two numbers is their mean: add them and divide by 2.(−2 + 4) ÷ 2 = 2 ÷ 2 = 1
- Find the y-coordinate of the midpoint. Do exactly the same with the y-coordinates.(5 + (−1)) ÷ 2 = 4 ÷ 2 = 2
- Write the answer as a coordinate pair. Across first, then up or down.Midpoint = (1, 2)
Check. From A to the midpoint the x-coordinate changes by 3 and from the midpoint to B it changes by 3; the y-coordinate changes by −3 and −3. The two steps are equal, so M is halfway. ✓
Watch out. Add the coordinates and halve. Subtracting gives the length of the gap, not the halfway point. Work with x and y separately.
Key words. The midpoint of a line segment is the point exactly halfway along it. The mean of two numbers is their sum divided by 2.
Method. The midpoint is exactly halfway between two points. Find the mean of the x-coordinates and the mean of the y-coordinates.
- Write down the two points and pick out their coordinates. The midpoint uses the x-coordinates together and the y-coordinates together.A = (−8, −6): x1 = −8, y1 = −6
B = (−2, −4): x2 = −2, y2 = −4 - Find the x-coordinate of the midpoint. The number halfway between two numbers is their mean: add them and divide by 2.((−8) + (−2)) ÷ 2 = −10 ÷ 2 = −5
- Find the y-coordinate of the midpoint. Do exactly the same with the y-coordinates.((−6) + (−4)) ÷ 2 = −10 ÷ 2 = −5
- Write the answer as a coordinate pair. Across first, then up or down.Midpoint = (−5, −5)
Check. From A to the midpoint the x-coordinate changes by 3 and from the midpoint to B it changes by 3; the y-coordinate changes by 1 and 1. The two steps are equal, so M is halfway. ✓
Watch out. Add the coordinates and halve. Subtracting gives the length of the gap, not the halfway point. Work with x and y separately.
Key words. The midpoint of a line segment is the point exactly halfway along it. The mean of two numbers is their sum divided by 2.
Method. The midpoint is exactly halfway between two points. Find the mean of the x-coordinates and the mean of the y-coordinates.
- Write down the two points and pick out their coordinates. The midpoint uses the x-coordinates together and the y-coordinates together.A = (1, 2): x1 = 1, y1 = 2
B = (6, 5): x2 = 6, y2 = 5 - Find the x-coordinate of the midpoint. The number halfway between two numbers is their mean: add them and divide by 2.(1 + 6) ÷ 2 = 7 ÷ 2 = 3.5
- Find the y-coordinate of the midpoint. Do exactly the same with the y-coordinates.(2 + 5) ÷ 2 = 7 ÷ 2 = 3.5
- Write the answer as a coordinate pair. Across first, then up or down.Midpoint = (3.5, 3.5)
Check. From A to the midpoint the x-coordinate changes by 2.5 and from the midpoint to B it changes by 2.5; the y-coordinate changes by 1.5 and 1.5. The two steps are equal, so M is halfway. ✓
Watch out. Add the coordinates and halve. Subtracting gives the length of the gap, not the halfway point. Work with x and y separately.
Key words. The midpoint of a line segment is the point exactly halfway along it. The mean of two numbers is their sum divided by 2.
Method. The midpoint is exactly halfway between two points. Find the mean of the x-coordinates and the mean of the y-coordinates.
- Write down the two points and pick out their coordinates. The midpoint uses the x-coordinates together and the y-coordinates together.A = (−3, 4): x1 = −3, y1 = 4
B = (2, −1): x2 = 2, y2 = −1 - Find the x-coordinate of the midpoint. The number halfway between two numbers is their mean: add them and divide by 2.(−3 + 2) ÷ 2 = −1 ÷ 2 = −0.5
- Find the y-coordinate of the midpoint. Do exactly the same with the y-coordinates.(4 + (−1)) ÷ 2 = 3 ÷ 2 = 1.5
- Write the answer as a coordinate pair. Across first, then up or down.Midpoint = (−0.5, 1.5)
Check. From A to the midpoint the x-coordinate changes by 2.5 and from the midpoint to B it changes by 2.5; the y-coordinate changes by −2.5 and −2.5. The two steps are equal, so M is halfway. ✓
Watch out. Add the coordinates and halve. Subtracting gives the length of the gap, not the halfway point. Work with x and y separately.
Key words. The midpoint of a line segment is the point exactly halfway along it. The mean of two numbers is their sum divided by 2.
Method. The midpoint is exactly halfway between two points. Find the mean of the x-coordinates and the mean of the y-coordinates.
- Write down the two points and pick out their coordinates. The midpoint uses the x-coordinates together and the y-coordinates together.A = (0, −5): x1 = 0, y1 = −5
B = (7, 0): x2 = 7, y2 = 0 - Find the x-coordinate of the midpoint. The number halfway between two numbers is their mean: add them and divide by 2.(0 + 7) ÷ 2 = 7 ÷ 2 = 3.5
- Find the y-coordinate of the midpoint. Do exactly the same with the y-coordinates.(−5 + 0) ÷ 2 = −5 ÷ 2 = −2.5
- Write the answer as a coordinate pair. Across first, then up or down.Midpoint = (3.5, −2.5)
Check. From A to the midpoint the x-coordinate changes by 3.5 and from the midpoint to B it changes by 3.5; the y-coordinate changes by 2.5 and 2.5. The two steps are equal, so M is halfway. ✓
Watch out. Add the coordinates and halve. Subtracting gives the length of the gap, not the halfway point. Work with x and y separately.
Key words. The midpoint of a line segment is the point exactly halfway along it. The mean of two numbers is their sum divided by 2.
Method. The midpoint is exactly halfway between two points. Find the mean of the x-coordinates and the mean of the y-coordinates.
- Write down the two points and pick out their coordinates. The midpoint uses the x-coordinates together and the y-coordinates together.A = (−1, 3): x1 = −1, y1 = 3
B = (0, 6): x2 = 0, y2 = 6 - Find the x-coordinate of the midpoint. The number halfway between two numbers is their mean: add them and divide by 2.(−1 + 0) ÷ 2 = −1 ÷ 2 = −0.5
- Find the y-coordinate of the midpoint. Do exactly the same with the y-coordinates.(3 + 6) ÷ 2 = 9 ÷ 2 = 4.5
- Write the answer as a coordinate pair. Across first, then up or down.Midpoint = (−0.5, 4.5)
Check. From A to the midpoint the x-coordinate changes by 0.5 and from the midpoint to B it changes by 0.5; the y-coordinate changes by 1.5 and 1.5. The two steps are equal, so M is halfway. ✓
Watch out. Add the coordinates and halve. Subtracting gives the length of the gap, not the halfway point. Work with x and y separately.
Key words. The midpoint of a line segment is the point exactly halfway along it. The mean of two numbers is their sum divided by 2.
Method. The midpoint is exactly halfway between two points. Find the mean of the x-coordinates and the mean of the y-coordinates.
- Write down the two points and pick out their coordinates. The midpoint uses the x-coordinates together and the y-coordinates together.A = (−6, 2): x1 = −6, y1 = 2
B = (4, 8): x2 = 4, y2 = 8 - Find the x-coordinate of the midpoint. The number halfway between two numbers is their mean: add them and divide by 2.(−6 + 4) ÷ 2 = −2 ÷ 2 = −1
- Find the y-coordinate of the midpoint. Do exactly the same with the y-coordinates.(2 + 8) ÷ 2 = 10 ÷ 2 = 5
- Write the answer as a coordinate pair. Across first, then up or down.Midpoint = (−1, 5)
- Sketch to see that it makes sense. The midpoint M must be on the line, exactly halfway between the two ends.
Check. From A to the midpoint the x-coordinate changes by 5 and from the midpoint to B it changes by 5; the y-coordinate changes by 3 and 3. The two steps are equal, so M is halfway. ✓
Watch out. Add the coordinates and halve. Subtracting gives the length of the gap, not the halfway point. Work with x and y separately.
Key words. The midpoint of a line segment is the point exactly halfway along it. The mean of two numbers is their sum divided by 2.
Method. The midpoint is exactly halfway between two points. Find the mean of the x-coordinates and the mean of the y-coordinates.
- Write down the two points and pick out their coordinates. The midpoint uses the x-coordinates together and the y-coordinates together.A = (−5, 2): x1 = −5, y1 = 2
B = (3, −4): x2 = 3, y2 = −4 - Find the x-coordinate of the midpoint. The number halfway between two numbers is their mean: add them and divide by 2.(−5 + 3) ÷ 2 = −2 ÷ 2 = −1
- Find the y-coordinate of the midpoint. Do exactly the same with the y-coordinates.(2 + (−4)) ÷ 2 = −2 ÷ 2 = −1
- Write the answer as a coordinate pair. Across first, then up or down.Midpoint = (−1, −1)
- Sketch to see that it makes sense. The midpoint M must be on the line, exactly halfway between the two ends.
Check. From A to the midpoint the x-coordinate changes by 4 and from the midpoint to B it changes by 4; the y-coordinate changes by −3 and −3. The two steps are equal, so M is halfway. ✓
Watch out. Add the coordinates and halve. Subtracting gives the length of the gap, not the halfway point. Work with x and y separately.
Method. Check Tom’s method against the rule: the midpoint adds the coordinates and halves.
- Look at what Tom did. He worked out (8 − 2) ÷ 2 = 3 and (−2 − 6) ÷ 2 = −4. He subtracted instead of adding.Tom: (3, −4)
- Do it correctly: add the x-coordinates and halve. Midpoint uses the mean of the coordinates.(2 + 8) ÷ 2 = 10 ÷ 2 = 5
- Add the y-coordinates and halve. (6 + (−2)) ÷ 2 = 4 ÷ 2 = 2
- Write the correct midpoint. (5, 2)
Check. (5, 2) is 3 right of 2 and 3 left of 8, and 4 below 6 and 4 above −2. ✓
Watch out. Subtracting gives half the distance between the points, not the halfway point.
Method. Use the midpoint rule twice: first for C (the midpoint of AB), then for D (the midpoint of AC).
- Find C, the midpoint of A and B. Add the coordinates and halve.x: (−4 + 8) ÷ 2 = 2
y: (6 + (−2)) ÷ 2 = 2
C = (2, 2) - Find D, the midpoint of A and C. Now use A and the new point C.x: (−4 + 2) ÷ 2 = −1
y: (6 + 2) ÷ 2 = 4
D = (−1, 4)
Check. D should be a quarter of the way from A to B. A to B is 12 across and −8 up, and a quarter of that is 3 across and −2 up. A + that = (−1, 4). ✓
Watch out. D is the midpoint of AC, not of AB.
Key words. The midpoint of a line segment is the point exactly halfway along it. The mean of two numbers is their sum divided by 2.
Method. The midpoint is exactly halfway between two points. Find the mean of the x-coordinates and the mean of the y-coordinates.
- Write down the two points and pick out their coordinates. The midpoint uses the x-coordinates together and the y-coordinates together.A = (−1, −2): x1 = −1, y1 = −2
C = (5, 4): x2 = 5, y2 = 4 - Find the x-coordinate of the midpoint. The number halfway between two numbers is their mean: add them and divide by 2.(−1 + 5) ÷ 2 = 4 ÷ 2 = 2
- Find the y-coordinate of the midpoint. Do exactly the same with the y-coordinates.(−2 + 4) ÷ 2 = 2 ÷ 2 = 1
- Write the answer as a coordinate pair. Across first, then up or down.Midpoint = (2, 1)
Check. From A to the midpoint the x-coordinate changes by 3 and from the midpoint to C it changes by 3; the y-coordinate changes by 3 and 3. The two steps are equal, so M is halfway. ✓
Watch out. Add the coordinates and halve. Subtracting gives the length of the gap, not the halfway point. Work with x and y separately.
Key words. The midpoint of a line segment is the point exactly halfway along it. The mean of two numbers is their sum divided by 2.
Method. The midpoint is exactly halfway between two points. Find the mean of the x-coordinates and the mean of the y-coordinates.
- Write down the two points and pick out their coordinates. The midpoint uses the x-coordinates together and the y-coordinates together.B = (5, −2): x1 = 5, y1 = −2
D = (−1, 4): x2 = −1, y2 = 4 - Find the x-coordinate of the midpoint. The number halfway between two numbers is their mean: add them and divide by 2.(5 + (−1)) ÷ 2 = 4 ÷ 2 = 2
- Find the y-coordinate of the midpoint. Do exactly the same with the y-coordinates.(−2 + 4) ÷ 2 = 2 ÷ 2 = 1
- Write the answer as a coordinate pair. Across first, then up or down.Midpoint = (2, 1)
Check. From B to the midpoint the x-coordinate changes by −3 and from the midpoint to D it changes by −3; the y-coordinate changes by 3 and 3. The two steps are equal, so M is halfway. ✓
Watch out. Add the coordinates and halve. Subtracting gives the length of the gap, not the halfway point. Work with x and y separately.
Key words. To bisect means to cut exactly in half. A diagonal joins opposite corners.
Method. Compare the two midpoints.
- Compare. Both midpoints are (2, 1).midpoint of AC = (2, 1), midpoint of BD = (2, 1)
- Say what it means. If two lines share a midpoint, each cuts the other in half (they bisect each other).The diagonals of a rectangle cross at their common midpoint.
Check. In every rectangle the diagonals cross at the centre. ✓
Key words. The midpoint of a line segment is the point exactly halfway along it. The mean of two numbers is their sum divided by 2.
Method. The midpoint is exactly halfway between two points. Find the mean of the x-coordinates and the mean of the y-coordinates.
- Write down the two points and pick out their coordinates. The midpoint uses the x-coordinates together and the y-coordinates together.C = (2, −3): x1 = 2, y1 = −3
L = (10, 5): x2 = 10, y2 = 5 - Find the x-coordinate of the midpoint. The number halfway between two numbers is their mean: add them and divide by 2.(2 + 10) ÷ 2 = 12 ÷ 2 = 6
- Find the y-coordinate of the midpoint. Do exactly the same with the y-coordinates.(−3 + 5) ÷ 2 = 2 ÷ 2 = 1
- Write the answer as a coordinate pair. Across first, then up or down.Midpoint = (6, 1)
Check. From C to the midpoint the x-coordinate changes by 4 and from the midpoint to L it changes by 4; the y-coordinate changes by 4 and 4. The two steps are equal, so M is halfway. ✓
Watch out. Add the coordinates and halve. Subtracting gives the length of the gap, not the halfway point. Work with x and y separately.
Key words. The midpoint of a line segment is the point exactly halfway along it. The mean of two numbers is their sum divided by 2.
Method. The midpoint is exactly halfway between two points. Find the mean of the x-coordinates and the mean of the y-coordinates.
- Write down the two points and pick out their coordinates. The midpoint uses the x-coordinates together and the y-coordinates together.B = (6, 1): x1 = 6, y1 = 1
L = (10, 5): x2 = 10, y2 = 5 - Find the x-coordinate of the midpoint. The number halfway between two numbers is their mean: add them and divide by 2.(6 + 10) ÷ 2 = 16 ÷ 2 = 8
- Find the y-coordinate of the midpoint. Do exactly the same with the y-coordinates.(1 + 5) ÷ 2 = 6 ÷ 2 = 3
- Write the answer as a coordinate pair. Across first, then up or down.Midpoint = (8, 3)
Check. From B to the midpoint the x-coordinate changes by 2 and from the midpoint to L it changes by 2; the y-coordinate changes by 2 and 2. The two steps are equal, so M is halfway. ✓
Watch out. Add the coordinates and halve. Subtracting gives the length of the gap, not the halfway point. Work with x and y separately.
Exercise 12C Lengths of line segments
Calculator allowedKey words. A line segment is the part of a line between two points. Pythagoras’ theorem says that in a right-angled triangle the square of the hypotenuse equals the sum of the squares of the other two sides (see Chapter 20).
Method. Draw a right-angled triangle using the horizontal and vertical distances between the points, then use Pythagoras’ theorem to find the length of the line (the hypotenuse).
- Find the horizontal distance between the points (the difference in x). Subtract the x-coordinates and ignore the sign: a distance cannot be negative.3 − 0 = 3
- Find the vertical distance (the difference in y). Do the same with the y-coordinates.4 − 0 = 4
- Sketch a right-angled triangle on the line. The line AB is the hypotenuse (the longest side, opposite the right angle). The two shorter sides are the horizontal and vertical distances.
- Use Pythagoras’ theorem: c² = a² + b². Square the two shorter sides and add them to get the hypotenuse squared.AB² = 3² + 4² = 9 + 16 = 25
- Take the square root. The square root undoes the squaring, giving the length itself.AB = √25 = 5
Check. 5² = 25 = 25. ✓
Watch out. Add the squares before square-rooting. √(9 + 16) is 5, not 3 + 4.
Key words. A line segment is the part of a line between two points. Pythagoras’ theorem says that in a right-angled triangle the square of the hypotenuse equals the sum of the squares of the other two sides (see Chapter 20).
Method. Draw a right-angled triangle using the horizontal and vertical distances between the points, then use Pythagoras’ theorem to find the length of the line (the hypotenuse).
- Find the horizontal distance between the points (the difference in x). Subtract the x-coordinates and ignore the sign: a distance cannot be negative.6 − 1 = 5
- Find the vertical distance (the difference in y). Do the same with the y-coordinates.13 − 1 = 12
- Sketch a right-angled triangle on the line. The line AB is the hypotenuse (the longest side, opposite the right angle). The two shorter sides are the horizontal and vertical distances.
- Use Pythagoras’ theorem: c² = a² + b². Square the two shorter sides and add them to get the hypotenuse squared.AB² = 5² + 12² = 25 + 144 = 169
- Take the square root. The square root undoes the squaring, giving the length itself.AB = √169 = 13
Check. 13² = 169 = 169. ✓
Watch out. Add the squares before square-rooting. √(9 + 16) is 5, not 3 + 4.
Key words. A line segment is the part of a line between two points. Pythagoras’ theorem says that in a right-angled triangle the square of the hypotenuse equals the sum of the squares of the other two sides (see Chapter 20).
Method. Draw a right-angled triangle using the horizontal and vertical distances between the points, then use Pythagoras’ theorem to find the length of the line (the hypotenuse).
- Find the horizontal distance between the points (the difference in x). Subtract the x-coordinates and ignore the sign: a distance cannot be negative.4 − (−2) = 6
- Find the vertical distance (the difference in y). Do the same with the y-coordinates.7 − (−1) = 8
- Sketch a right-angled triangle on the line. The line AB is the hypotenuse (the longest side, opposite the right angle). The two shorter sides are the horizontal and vertical distances.
- Use Pythagoras’ theorem: c² = a² + b². Square the two shorter sides and add them to get the hypotenuse squared.AB² = 6² + 8² = 36 + 64 = 100
- Take the square root. The square root undoes the squaring, giving the length itself.AB = √100 = 10
Check. 10² = 100 = 100. ✓
Watch out. Add the squares before square-rooting. √(9 + 16) is 5, not 3 + 4.
Key words. A line segment is the part of a line between two points. Pythagoras’ theorem says that in a right-angled triangle the square of the hypotenuse equals the sum of the squares of the other two sides (see Chapter 20).
Method. Draw a right-angled triangle using the horizontal and vertical distances between the points, then use Pythagoras’ theorem to find the length of the line (the hypotenuse).
- Find the horizontal distance between the points (the difference in x). Subtract the x-coordinates and ignore the sign: a distance cannot be negative.2 − (−3) = 5
- Find the vertical distance (the difference in y). Do the same with the y-coordinates.14 − 2 = 12
- Sketch a right-angled triangle on the line. The line AB is the hypotenuse (the longest side, opposite the right angle). The two shorter sides are the horizontal and vertical distances.
- Use Pythagoras’ theorem: c² = a² + b². Square the two shorter sides and add them to get the hypotenuse squared.AB² = 5² + 12² = 25 + 144 = 169
- Take the square root. The square root undoes the squaring, giving the length itself.AB = √169 = 13
Check. 13² = 169 = 169. ✓
Watch out. Add the squares before square-rooting. √(9 + 16) is 5, not 3 + 4.
Method. The horizontal distance is the difference in x-coordinates.
- Take the x-coordinates. O has x = 0 and P has x = 6.0 and 6
- Subtract. Bigger minus smaller.6 − 0 = 6
Check. P is 6 squares to the right of O. ✓
Method. The vertical distance is the difference in y-coordinates.
- Take the y-coordinates. O has y = 0 and P has y = 8.0 and 8
- Subtract. Bigger minus smaller.8 − 0 = 8
Check. P is 8 squares above O. ✓
Key words. A line segment is the part of a line between two points. Pythagoras’ theorem says that in a right-angled triangle the square of the hypotenuse equals the sum of the squares of the other two sides (see Chapter 20).
Method. Draw a right-angled triangle using the horizontal and vertical distances between the points, then use Pythagoras’ theorem to find the length of the line (the hypotenuse).
- Find the horizontal distance between the points (the difference in x). Subtract the x-coordinates and ignore the sign: a distance cannot be negative.6 − 0 = 6
- Find the vertical distance (the difference in y). Do the same with the y-coordinates.8 − 0 = 8
- Sketch a right-angled triangle on the line. The line OP is the hypotenuse (the longest side, opposite the right angle). The two shorter sides are the horizontal and vertical distances.
- Use Pythagoras’ theorem: c² = a² + b². Square the two shorter sides and add them to get the hypotenuse squared.OP² = 6² + 8² = 36 + 64 = 100
- Take the square root. The square root undoes the squaring, giving the length itself.OP = √100 = 10
Check. 10² = 100 = 100. ✓
Watch out. Add the squares before square-rooting. √(9 + 16) is 5, not 3 + 4.
Key words. A line segment is the part of a line between two points. Pythagoras’ theorem says that in a right-angled triangle the square of the hypotenuse equals the sum of the squares of the other two sides (see Chapter 20).
Method. Draw a right-angled triangle using the horizontal and vertical distances between the points, then use Pythagoras’ theorem to find the length of the line (the hypotenuse).
- Find the horizontal distance between the points (the difference in x). Subtract the x-coordinates and ignore the sign: a distance cannot be negative.4 − (−1) = 5
- Find the vertical distance (the difference in y). Do the same with the y-coordinates.5 − 3 = 2
- Sketch a right-angled triangle on the line. The line AB is the hypotenuse (the longest side, opposite the right angle). The two shorter sides are the horizontal and vertical distances.
- Use Pythagoras’ theorem: c² = a² + b². Square the two shorter sides and add them to get the hypotenuse squared.AB² = 5² + 2² = 25 + 4 = 29
- Take the square root and round. The square root undoes the squaring. Use a calculator, then round as the question asks.AB = √29 = 5.3852...
= 5.4 to 1 decimal place
Check. The line should be longer than the longer leg (5) and shorter than the two legs added (7). 5.4 is in between. ✓
Watch out. Add the squares before square-rooting. √(9 + 16) is 5, not 3 + 4.
Key words. A line segment is the part of a line between two points. Pythagoras’ theorem says that in a right-angled triangle the square of the hypotenuse equals the sum of the squares of the other two sides (see Chapter 20).
Method. Draw a right-angled triangle using the horizontal and vertical distances between the points, then use Pythagoras’ theorem to find the length of the line (the hypotenuse).
- Find the horizontal distance between the points (the difference in x). Subtract the x-coordinates and ignore the sign: a distance cannot be negative.2 − (−3) = 5
- Find the vertical distance (the difference in y). Do the same with the y-coordinates.4 − (−2) = 6
- Sketch a right-angled triangle on the line. The line AB is the hypotenuse (the longest side, opposite the right angle). The two shorter sides are the horizontal and vertical distances.
- Use Pythagoras’ theorem: c² = a² + b². Square the two shorter sides and add them to get the hypotenuse squared.AB² = 5² + 6² = 25 + 36 = 61
- Take the square root and round. The square root undoes the squaring. Use a calculator, then round as the question asks.AB = √61 = 7.8102...
= 7.8 to 1 decimal place
Check. The line should be longer than the longer leg (6) and shorter than the two legs added (11). 7.8 is in between. ✓
Watch out. Add the squares before square-rooting. √(9 + 16) is 5, not 3 + 4.
Key words. A line segment is the part of a line between two points. Pythagoras’ theorem says that in a right-angled triangle the square of the hypotenuse equals the sum of the squares of the other two sides (see Chapter 20).
Method. Draw a right-angled triangle using the horizontal and vertical distances between the points, then use Pythagoras’ theorem to find the length of the line (the hypotenuse).
- Find the horizontal distance between the points (the difference in x). Subtract the x-coordinates and ignore the sign: a distance cannot be negative.7 − 2 = 5
- Find the vertical distance (the difference in y). Do the same with the y-coordinates.5 − (−3) = 8
- Sketch a right-angled triangle on the line. The line AB is the hypotenuse (the longest side, opposite the right angle). The two shorter sides are the horizontal and vertical distances.
- Use Pythagoras’ theorem: c² = a² + b². Square the two shorter sides and add them to get the hypotenuse squared.AB² = 5² + 8² = 25 + 64 = 89
- Take the square root and round. The square root undoes the squaring. Use a calculator, then round as the question asks.AB = √89 = 9.4340...
= 9.4 to 1 decimal place
Check. The line should be longer than the longer leg (8) and shorter than the two legs added (13). 9.4 is in between. ✓
Watch out. Add the squares before square-rooting. √(9 + 16) is 5, not 3 + 4.
Key words. A line segment is the part of a line between two points. Pythagoras’ theorem says that in a right-angled triangle the square of the hypotenuse equals the sum of the squares of the other two sides (see Chapter 20).
Method. Draw a right-angled triangle using the horizontal and vertical distances between the points, then use Pythagoras’ theorem to find the length of the line (the hypotenuse).
- Find the horizontal distance between the points (the difference in x). Subtract the x-coordinates and ignore the sign: a distance cannot be negative.−1 − (−5) = 4
- Find the vertical distance (the difference in y). Do the same with the y-coordinates.1 − (−6) = 7
- Sketch a right-angled triangle on the line. The line AB is the hypotenuse (the longest side, opposite the right angle). The two shorter sides are the horizontal and vertical distances.
- Use Pythagoras’ theorem: c² = a² + b². Square the two shorter sides and add them to get the hypotenuse squared.AB² = 4² + 7² = 16 + 49 = 65
- Take the square root and round. The square root undoes the squaring. Use a calculator, then round as the question asks.AB = √65 = 8.0623...
= 8.1 to 1 decimal place
Check. The line should be longer than the longer leg (7) and shorter than the two legs added (11). 8.1 is in between. ✓
Watch out. Add the squares before square-rooting. √(9 + 16) is 5, not 3 + 4.
Key words. A line segment is the part of a line between two points. Pythagoras’ theorem says that in a right-angled triangle the square of the hypotenuse equals the sum of the squares of the other two sides (see Chapter 20).
Method. Draw a right-angled triangle using the horizontal and vertical distances between the points, then use Pythagoras’ theorem to find the length of the line (the hypotenuse).
- Find the horizontal distance between the points (the difference in x). Subtract the x-coordinates and ignore the sign: a distance cannot be negative.5 − 0 = 5
- Find the vertical distance (the difference in y). Do the same with the y-coordinates.12 − 0 = 12
- Sketch a right-angled triangle on the line. The line HL is the hypotenuse (the longest side, opposite the right angle). The two shorter sides are the horizontal and vertical distances.
- Use Pythagoras’ theorem: c² = a² + b². Square the two shorter sides and add them to get the hypotenuse squared.HL² = 5² + 12² = 25 + 144 = 169
- Take the square root. The square root undoes the squaring, giving the length itself.HL = √169 = 13 km
Check. 13² = 169 = 169. ✓
Watch out. Add the squares before square-rooting. √(9 + 16) is 5, not 3 + 4.
Key words. A line segment is the part of a line between two points. Pythagoras’ theorem says that in a right-angled triangle the square of the hypotenuse equals the sum of the squares of the other two sides (see Chapter 20).
Method. Draw a right-angled triangle using the horizontal and vertical distances between the points, then use Pythagoras’ theorem to find the length of the line (the hypotenuse).
- Find the horizontal distance between the points (the difference in x). Subtract the x-coordinates and ignore the sign: a distance cannot be negative.5 − (−4) = 9
- Find the vertical distance (the difference in y). Do the same with the y-coordinates.12 − 0 = 12
- Sketch a right-angled triangle on the line. The line LB is the hypotenuse (the longest side, opposite the right angle). The two shorter sides are the horizontal and vertical distances.
- Use Pythagoras’ theorem: c² = a² + b². Square the two shorter sides and add them to get the hypotenuse squared.LB² = 9² + 12² = 81 + 144 = 225
- Take the square root. The square root undoes the squaring, giving the length itself.LB = √225 = 15 km
Check. 15² = 225 = 225. ✓
Watch out. Add the squares before square-rooting. √(9 + 16) is 5, not 3 + 4.
Method. Add the two distances found in parts (a) and (b).
- Write the two distances. The trip has two parts, HL and then LB.HL = 13 km, LB = 15 km
- Add them. 13 + 15 = 28 km
Check. The straight route HB is only 4 km, but the trip goes round the lighthouse, so it is much longer. ✓
Method. Find the length of each side, then add them. Two sides are parallel to the axes, so only one needs Pythagoras.
- Draw a sketch. The sketch shows a right angle at (7, 1).
- Side AB (horizontal). Same y-coordinate, so subtract the x-coordinates.7 − 1 = 6
- Side BC (vertical). Same x-coordinate, so subtract the y-coordinates.9 − 1 = 8
- Side AC (sloping). This is the hypotenuse. Use Pythagoras.AC² = 6² + 8² = 36 + 64 = 100, so AC = √100 = 10
- Add the three sides. Perimeter is the total distance round.6 + 8 + 10 = 24
Check. 6, 8, 10 is a Pythagorean triple (3, 4, 5 doubled), and 6 + 8 + 10 = 24. ✓
Watch out. Do not forget the sloping side when you add the perimeter.
Method. Work out how far each point is from the origin using Pythagoras. You can compare the squares of the distances without taking square roots.
- Distance² of (3, −5) from (0, 0). Square both coordinates and add. The minus sign disappears when you square.3² + (−5)² = 9 + 25 = 34
- Distance² of (−4, 4) from (0, 0). (−4)² + 4² = 16 + 16 = 32
- Compare. The point with the bigger distance² is further away, because distances are positive.34 > 32, so (3, −5) is further.
- Optional: the actual distances. √34 = 5.83... and √32 = 5.66...
Check. 5.83 > 5.66, which agrees. ✓
Watch out. (−5)² = +25, not −25. A negative number squared is positive.
Method. The 12 across and 5 up form a right-angled triangle. Find the length in grid units, then convert to metres.
- Horizontal distance. Subtract the x-coordinates.10 − (−2) = 12
- Vertical distance. Subtract the y-coordinates.8 − 3 = 5
- Use Pythagoras. 12, 5, 13 is a Pythagorean triple.length² = 12² + 5² = 144 + 25 = 169, so length = √169 = 13 units
- Convert to metres. Each unit is 100 m.13 × 100 = 1300 m
Check. 1300 m is 1.3 km, which is sensible for a drone flight across a park. ✓
Watch out. Convert units at the end: the grid length 13 is in grid units, not metres.
Key words. A line segment is the part of a line between two points. Pythagoras’ theorem says that in a right-angled triangle the square of the hypotenuse equals the sum of the squares of the other two sides (see Chapter 20).
Method. Draw a right-angled triangle using the horizontal and vertical distances between the points, then use Pythagoras’ theorem to find the length of the line (the hypotenuse).
- Find the horizontal distance between the points (the difference in x). Subtract the x-coordinates and ignore the sign: a distance cannot be negative.6 − (−6) = 12
- Find the vertical distance (the difference in y). Do the same with the y-coordinates.3 − (−2) = 5
- Sketch a right-angled triangle on the line. The line AB is the hypotenuse (the longest side, opposite the right angle). The two shorter sides are the horizontal and vertical distances.
- Use Pythagoras’ theorem: c² = a² + b². Square the two shorter sides and add them to get the hypotenuse squared.AB² = 12² + 5² = 144 + 25 = 169
- Take the square root. The square root undoes the squaring, giving the length itself.AB = √169 = 13 km
Check. 13² = 169 = 169. ✓
Watch out. Add the squares before square-rooting. √(9 + 16) is 5, not 3 + 4.
Key words. The midpoint of a line segment is the point exactly halfway along it. The mean of two numbers is their sum divided by 2.
Method. The midpoint is exactly halfway between two points. Find the mean of the x-coordinates and the mean of the y-coordinates.
- Write down the two points and pick out their coordinates. The midpoint uses the x-coordinates together and the y-coordinates together.A = (−6, −2): x1 = −6, y1 = −2
B = (6, 3): x2 = 6, y2 = 3 - Find the x-coordinate of the midpoint. The number halfway between two numbers is their mean: add them and divide by 2.(−6 + 6) ÷ 2 = 0 ÷ 2 = 0
- Find the y-coordinate of the midpoint. Do exactly the same with the y-coordinates.(−2 + 3) ÷ 2 = 1 ÷ 2 = 0.5
- Write the answer as a coordinate pair. Across first, then up or down.Midpoint = (0, 0.5)
Check. From A to the midpoint the x-coordinate changes by 6 and from the midpoint to B it changes by 6; the y-coordinate changes by 2.5 and 2.5. The two steps are equal, so M is halfway. ✓
Watch out. Add the coordinates and halve. Subtracting gives the length of the gap, not the halfway point. Work with x and y separately.
Method. Check what Mia did against Pythagoras’ theorem.
- Find the horizontal and vertical distances. Subtract the coordinates.4 − 1 = 3 across and 6 − 2 = 4 up
- See what Mia did. She added 3 and 4 to get 7. But adding the sides of the triangle gives the length of the two short sides together, not the length of the sloping line.Mia: 3 + 4 = 7
- Do it correctly with Pythagoras. Square, add, then square root.length² = 3² + 4² = 9 + 16 = 25, so length = √25 = 5
- Compare. A straight line is shorter than going round two sides of a triangle.5 < 7, so Mia is wrong.
Check. A 3, 4, 5 triangle is the best-known right-angled triangle. ✓
Watch out. The straight line between two points is always shorter than going across and then up.
Method. Use the horizontal and vertical distances and Pythagoras.
- Horizontal distance. Subtract the x-coordinates.4 − (−4) = 8
- Vertical distance. Subtract the y-coordinates.3 − (−3) = 6
- Pythagoras. 8, 6, 10 is a 4, 3, 5 triangle doubled.length² = 8² + 6² = 64 + 36 = 100
- Square root. length = √100 = 10 km
Check. 8, 6, 10 is the 3, 4, 5 triangle with every side doubled (6, 8, 10). ✓
Watch out. The diagonal is shorter than 8 + 6 = 14 km, as it should be.