GCSE Maths: Midpoints and lengths of line segments (Higher)

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.

1. 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
2. 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
3. Find the y-coordinate of the midpoint. Do exactly the same with the y-coordinates.
   (4 + 10) ÷ 2 = 14 ÷ 2 = 7
4. 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.

1. 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
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
3. Find the y-coordinate of the midpoint. Do exactly the same with the y-coordinates.
   (6 + 2) ÷ 2 = 8 ÷ 2 = 4
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.

1. 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
2. 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
3. Find the y-coordinate of the midpoint. Do exactly the same with the y-coordinates.
   (3 + 7) ÷ 2 = 10 ÷ 2 = 5
4. 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.

1. 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
2. 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
3. Find the y-coordinate of the midpoint. Do exactly the same with the y-coordinates.
   (2 + 8) ÷ 2 = 10 ÷ 2 = 5
4. 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.

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.

1. 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
2. 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
3. Find the y-coordinate of the midpoint. Do exactly the same with the y-coordinates.
   (2 + 8) ÷ 2 = 10 ÷ 2 = 5
4. 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.

1. 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
2. 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
3. Find the y-coordinate of the midpoint. Do exactly the same with the y-coordinates.
   (−3 + 5) ÷ 2 = 2 ÷ 2 = 1
4. 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.

1. 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
2. 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
3. Find the y-coordinate of the midpoint. Do exactly the same with the y-coordinates.
   (5 + (−1)) ÷ 2 = 4 ÷ 2 = 2
4. 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.

1. 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
2. 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
3. Find the y-coordinate of the midpoint. Do exactly the same with the y-coordinates.
   ((−6) + (−4)) ÷ 2 = −10 ÷ 2 = −5
4. 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.

1. 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
2. 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
3. Find the y-coordinate of the midpoint. Do exactly the same with the y-coordinates.
   (2 + 5) ÷ 2 = 7 ÷ 2 = 3.5
4. 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.

1. 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
2. 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
3. Find the y-coordinate of the midpoint. Do exactly the same with the y-coordinates.
   (4 + (−1)) ÷ 2 = 3 ÷ 2 = 1.5
4. 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.

1. 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
2. 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
3. Find the y-coordinate of the midpoint. Do exactly the same with the y-coordinates.
   (−5 + 0) ÷ 2 = −5 ÷ 2 = −2.5
4. 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.

1. 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
2. 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
3. Find the y-coordinate of the midpoint. Do exactly the same with the y-coordinates.
   (3 + 6) ÷ 2 = 9 ÷ 2 = 4.5
4. 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.

1. 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
2. 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
3. Find the y-coordinate of the midpoint. Do exactly the same with the y-coordinates.
   (−2 + 4) ÷ 2 = 2 ÷ 2 = 1
4. 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.

1. 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
2. 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
3. Find the y-coordinate of the midpoint. Do exactly the same with the y-coordinates.
   (−2 + 4) ÷ 2 = 2 ÷ 2 = 1
4. 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.

1. Compare. Both midpoints are (2, 1).
   midpoint of AC = (2, 1), midpoint of BD = (2, 1)
2. 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.

1. 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
2. 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
3. Find the y-coordinate of the midpoint. Do exactly the same with the y-coordinates.
   (−3 + 5) ÷ 2 = 2 ÷ 2 = 1
4. 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.

1. 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
2. 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
3. Find the y-coordinate of the midpoint. Do exactly the same with the y-coordinates.
   (1 + 5) ÷ 2 = 6 ÷ 2 = 3
4. 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.

Method. The midpoint is halfway, so the other end is the same distance beyond the midpoint. Alternatively use the midpoint rule backwards: 2 × midpoint − known end.

1. Write what we know. A is one end and M is the midpoint. The other end is B = (x, y), which we want to find.
   A = (1, 1), M = (4, 3)
2. Work out how far the midpoint is from the known end. The midpoint is halfway, so the other end is the same step again beyond M.
   Across: 4 − 1 = 3
   Up/down: 3 − 1 = 2
3. Add the same step again to the midpoint. Going from A to M is half of the journey to B, so repeat it once more starting at M.
   x: 4 + 3 = 7
   y: 3 + 2 = 5
4. Write the coordinates.
   B = (7, 5)

Check. Midpoint of (1, 1) and (7, 5): (1 + 7) ÷ 2 = 4 and (1 + 5) ÷ 2 = 3. This is M. ✓

Watch out. Do not just add or subtract the midpoint and the end. The step from the end to M must be repeated once more.

Method. The midpoint is halfway, so the other end is the same distance beyond the midpoint. Alternatively use the midpoint rule backwards: 2 × midpoint − known end.

1. Write what we know. A is one end and M is the midpoint. The other end is B = (x, y), which we want to find.
   A = (−2, 5), M = (1, 2)
2. Work out how far the midpoint is from the known end. The midpoint is halfway, so the other end is the same step again beyond M.
   Across: 1 − (−2) = 3
   Up/down: 2 − 5 = −3
3. Add the same step again to the midpoint. Going from A to M is half of the journey to B, so repeat it once more starting at M.
   x: 1 + 3 = 4
   y: 2 + (−3) = −1
4. Write the coordinates.
   B = (4, −1)

Check. Midpoint of (−2, 5) and (4, −1): (−2 + 4) ÷ 2 = 1 and (5 + (−1)) ÷ 2 = 2. This is M. ✓

Watch out. Do not just add or subtract the midpoint and the end. The step from the end to M must be repeated once more.

Method. The midpoint is halfway, so the other end is the same distance beyond the midpoint. Alternatively use the midpoint rule backwards: 2 × midpoint − known end.

1. Write what we know. A is one end and M is the midpoint. The other end is B = (x, y), which we want to find.
   A = (6, −3), M = (2, 0)
2. Work out how far the midpoint is from the known end. The midpoint is halfway, so the other end is the same step again beyond M.
   Across: 2 − 6 = −4
   Up/down: 0 − (−3) = 3
3. Add the same step again to the midpoint. Going from A to M is half of the journey to B, so repeat it once more starting at M.
   x: 2 + (−4) = −2
   y: 0 + 3 = 3
4. Write the coordinates.
   B = (−2, 3)

Check. Midpoint of (6, −3) and (−2, 3): (6 + (−2)) ÷ 2 = 2 and (−3 + 3) ÷ 2 = 0. This is M. ✓

Watch out. Do not just add or subtract the midpoint and the end. The step from the end to M must be repeated once more.

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).

1. 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
2. Find the vertical distance (the difference in y). Do the same with the y-coordinates.
   4 − 0 = 4
3. 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.
4. 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
5. 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).

1. 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
2. Find the vertical distance (the difference in y). Do the same with the y-coordinates.
   13 − 1 = 12
3. 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.
4. 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
5. 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).

1. 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
2. Find the vertical distance (the difference in y). Do the same with the y-coordinates.
   7 − (−1) = 8
3. 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.
4. 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
5. 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).

1. 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
2. Find the vertical distance (the difference in y). Do the same with the y-coordinates.
   14 − 2 = 12
3. 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.
4. 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
5. 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).

1. 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
2. Find the vertical distance (the difference in y). Do the same with the y-coordinates.
   5 − 3 = 2
3. 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.
4. 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
5. 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).

1. 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
2. Find the vertical distance (the difference in y). Do the same with the y-coordinates.
   4 − (−2) = 6
3. 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.
4. 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
5. 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).

1. 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
2. Find the vertical distance (the difference in y). Do the same with the y-coordinates.
   5 − (−3) = 8
3. 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.
4. 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
5. 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).

1. 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
2. Find the vertical distance (the difference in y). Do the same with the y-coordinates.
   1 − (−6) = 7
3. 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.
4. 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
5. 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).

1. 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
2. Find the vertical distance (the difference in y). Do the same with the y-coordinates.
   12 − 0 = 12
3. 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.
4. 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
5. 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).

1. 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
2. Find the vertical distance (the difference in y). Do the same with the y-coordinates.
   12 − 0 = 12
3. 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.
4. 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
5. 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).

1. Write the two distances. The trip has two parts, HL and then LB.
   HL = 13 km, LB = 15 km
2. 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. ✓

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).

1. 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
2. Find the vertical distance (the difference in y). Do the same with the y-coordinates.
   3 − (−2) = 5
3. 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.
4. 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
5. 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.

1. 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
2. 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
3. Find the y-coordinate of the midpoint. Do exactly the same with the y-coordinates.
   (−2 + 3) ÷ 2 = 1 ÷ 2 = 0.5
4. 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.

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).

1. 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
2. Find the vertical distance (the difference in y). Do the same with the y-coordinates.
   5 − 1 = 4
3. 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.
4. Use Pythagoras’ theorem: c² = a² + b². Square the two shorter sides and add them to get the hypotenuse squared.
   AB² = 6² + 4² = 36 + 16 = 52
5. Take the square root. This gives the exact length as a surd. A surd is a square root that is not a whole number.
   AB = √52
6. Simplify the surd. Look for a square number that divides 52: 52 = 4 × 13, and √4 = 2.
   √52 = √4 × √13 = 2√13

Check. On a calculator √52 = 7.211, which matches 2√13 = 7.211. A sketch agrees: the line is longer than 6 but shorter than 10. ✓

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).

1. 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 = 3
2. Find the vertical distance (the difference in y). Do the same with the y-coordinates.
   4 − (−3) = 7
3. 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.
4. Use Pythagoras’ theorem: c² = a² + b². Square the two shorter sides and add them to get the hypotenuse squared.
   AB² = 3² + 7² = 9 + 49 = 58
5. Take the square root. This gives the exact length as a surd. A surd is a square root that is not a whole number.
   AB = √58

Check. On a calculator √58 = 7.616, which matches √58 = 7.616. A sketch agrees: the line is longer than 7 but shorter than 10. ✓

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).

1. Find the horizontal distance between the points (the difference in x). Subtract the x-coordinates and ignore the sign: a distance cannot be negative.
   0 − (−6) = 6
2. Find the vertical distance (the difference in y). Do the same with the y-coordinates.
   0 − (−6) = 6
3. 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.
4. Use Pythagoras’ theorem: c² = a² + b². Square the two shorter sides and add them to get the hypotenuse squared.
   AB² = 6² + 6² = 36 + 36 = 72
5. Take the square root. This gives the exact length as a surd. A surd is a square root that is not a whole number.
   AB = √72
6. Simplify the surd. Look for a square number that divides 72: 72 = 36 × 2, and √36 = 6.
   √72 = √36 × √2 = 6√2

Check. On a calculator √72 = 8.485, which matches 6√2 = 8.485. A sketch agrees: the line is longer than 6 but shorter than 12. ✓

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).

1. 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 − (−1) = 6
2. Find the vertical distance (the difference in y). Do the same with the y-coordinates.
   2 − (−6) = 8
3. 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.
4. 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
5. Take the square root. The square root undoes the squaring.
   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).

1. 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 − 1 = 4
2. Find the vertical distance (the difference in y). Do the same with the y-coordinates.
   3 − 1 = 2
3. 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.
4. Use Pythagoras’ theorem: c² = a² + b². Square the two shorter sides and add them to get the hypotenuse squared.
   AB² = 4² + 2² = 16 + 4 = 20
5. Take the square root. This gives the exact length as a surd. A surd is a square root that is not a whole number.
   AB = √20
6. Simplify the surd. Look for a square number that divides 20: 20 = 4 × 5, and √4 = 2.
   √20 = √4 × √5 = 2√5

Check. On a calculator √20 = 4.472, which matches 2√5 = 4.472. A sketch agrees: the line is longer than 4 but shorter than 6. ✓

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).

1. 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 − 3 = 2
2. Find the vertical distance (the difference in y). Do the same with the y-coordinates.
   7 − 3 = 4
3. Sketch a right-angled triangle on the line. The line BC is the hypotenuse (the longest side, opposite the right angle). The two shorter sides are the horizontal and vertical distances.
4. Use Pythagoras’ theorem: c² = a² + b². Square the two shorter sides and add them to get the hypotenuse squared.
   BC² = 2² + 4² = 4 + 16 = 20
5. Take the square root. This gives the exact length as a surd. A surd is a square root that is not a whole number.
   BC = √20
6. Simplify the surd. Look for a square number that divides 20: 20 = 4 × 5, and √4 = 2.
   √20 = √4 × √5 = 2√5

Check. On a calculator √20 = 4.472, which matches 2√5 = 4.472. A sketch agrees: the line is longer than 4 but shorter than 6. ✓

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).

1. 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 − 1 = 2
2. Find the vertical distance (the difference in y). Do the same with the y-coordinates.
   7 − 1 = 6
3. Sketch a right-angled triangle on the line. The line AC is the hypotenuse (the longest side, opposite the right angle). The two shorter sides are the horizontal and vertical distances.
4. Use Pythagoras’ theorem: c² = a² + b². Square the two shorter sides and add them to get the hypotenuse squared.
   AC² = 2² + 6² = 4 + 36 = 40
5. Take the square root. This gives the exact length as a surd. A surd is a square root that is not a whole number.
   AC = √40
6. Simplify the surd. Look for a square number that divides 40: 40 = 4 × 10, and √4 = 2.
   √40 = √4 × √10 = 2√10

Check. On a calculator √40 = 6.325, which matches 2√10 = 6.325. A sketch agrees: the line is longer than 6 but shorter than 8. ✓

Watch out. Add the squares before square-rooting. √(9 + 16) is 5, not 3 + 4.

Method. Compare the lengths. Two equal sides make it isosceles; the converse of Pythagoras shows a right angle.

1. Compare AB and BC. Both were found in parts (a) and (b).
   AB = 2√5 and BC = 2√5, so AB = BC: the triangle is isosceles.
2. Square the three sides. Squaring a surd removes the root: (2√5)² = 4 × 5 = 20.
   AB² = 20, BC² = 20, AC² = 40
3. Test Pythagoras. If the square of the longest side equals the sum of the squares of the other two, the angle opposite the longest side is a right angle.
   AB² + BC² = 20 + 20 = 40 = AC²
4. Conclude. State the result in words.
   ABC is an isosceles right-angled triangle, with the right angle at B.

Check. In a right isosceles triangle the hypotenuse is √2 times a short side: 2√5 × √2 = 2√10. ✓

Watch out. Say what you have shown at the end: “isosceles” and “right-angled”.

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.

1. Write down the two points and pick out their coordinates. The midpoint uses the x-coordinates together and the y-coordinates together.
   A = (−3, 1): x₁ = −3, y₁ = 1
   B = (5, 7): x₂ = 5, y₂ = 7
2. Find the x-coordinate of the midpoint. The number halfway between two numbers is their mean: add them and divide by 2.
   (−3 + 5) ÷ 2 = 2 ÷ 2 = 1
3. Find the y-coordinate of the midpoint. Do exactly the same with the y-coordinates.
   (1 + 7) ÷ 2 = 8 ÷ 2 = 4
4. Write the answer as a coordinate pair. Across first, then up or down.
   Midpoint = (1, 4)

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.

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).

1. 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 − (−3) = 8
2. Find the vertical distance (the difference in y). Do the same with the y-coordinates.
   7 − 1 = 6
3. 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.
4. Use Pythagoras’ theorem: c² = a² + b². Square the two shorter sides and add them to get the hypotenuse squared.
   AB² = 8² + 6² = 64 + 36 = 100
5. 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.

Method. The radius is half the diameter.

1. Take the diameter from part (b).
   diameter = 10
2. Halve it. The radius goes from the centre to the edge, half way across.
   10 ÷ 2 = 5

Check. Distance from the centre (1, 4) to A(−3, 1): √(4² + 3²) = √25 = 5. ✓

Method. A point is on the circle if its distance from the centre equals the radius.

1. Write the centre and the point. The centre was found in part (a): (1, 4). The point is P(1, 9).
   C(1, 4), P(1, 9)
2. Find the distance CP. The x-coordinates are the same, so the line is vertical: subtract the y-coordinates.
   9 − 4 = 5
3. Compare with the radius. The radius is 5.
   CP = 5 = radius, so P is on the circle.

Check. Distance from the centre (1, 4) to P(1, 9) is 5 and the radius is half of AB = 10 ÷ 2 = 5. ✓

Watch out. Compare the distance from the centre, not from the end of the diameter.