Year 9 · Unit 2 Pythagoras, similarity and trigonometry · Lesson 4 of 7

Straight-line distance on a grid

Two points on a grid are the ends of the long side of a hidden right-angled triangle. Pythagoras gives the distance; averaging gives the midpoint.

Geometry and TrigonometryScientific calculator

Warm-up

Three quick ones from earlier work. Try each in your head or on paper before you open the answer.

  1. Work out 4 − (−3).
    Answer7: subtracting a negative adds.
  2. A right-angled triangle has legs 5 and 12. How long is the hypotenuse?
    Answer13, because 25 + 144 = 169.
  3. What is the mean of 6 and 15?
    Answer10.5, which is (6 + 15) ÷ 2.
Words used in this lesson
coordinates
a pair of numbers giving a position: across, then up. For example: (−2, 1).
distance
the length of the straight line between two points. For example: 10 m from start to chest.
midpoint
the point half-way between two points. For example: (−2, −1).
horizontal
level, from left to right. For example: the across step.
vertical
straight up and down. For example: the up step.

Along the grid, or straight across?

A game designer puts a treasure chest at (4, 9). A player starts at (−2, 1) on a grid where one square is 1 m. Walking along the grid lines the player goes 6 m across and 8 m up, which is 14 m. The straight line from the start to the chest is more than 14 m, exactly 14 m, or less than 14 m. Decide which, and then estimate it.

Distance: the steps across and up between two points are the legs of a right-angled triangle, so d² = (x2 − x1)² + (y2 − y1)² and d is the square root of that.

Midpoint: the average of the two x-coordinates and the average of the two y-coordinates: ((x1 + x2) ÷ 2, (y1 + y2) ÷ 2).

Move A and B around. Predict the distance first, then read it.

−3
1
2
4
√34 ≈ 5.83distance AB
(−0.5, 2.5)midpoint of AB

Across 5, up 3: d² = 25 + 9 = 34, so d = √34 ≈ 5.83. The midpoint is (−0.5, 2.5).

Show me the steps
  1. A(−3, 1) and B(2, 4): across 5, up 3. 25 + 9 = 34, so d = √34 ≈ 5.83. Midpoint (−0.5, 2.5).
  2. A(−4, −4) and B(1, 8): across 5, up 12. 25 + 144 = 169, so d = 13. Midpoint (−1.5, 2).
  3. A(1, 1) and B(4, 3): across 3, up 2. 9 + 4 = 13, so d = √13 ≈ 3.61. Midpoint (2.5, 2).
  4. A(−5, 2) and B(3, 2): across 8, up 0. The triangle has flattened, so d = 8. Midpoint (−1, 2).

Two slips to watch for. The step between −3 and 5 is 5 − (−3) = 8, not 2, so cross zero with care. And the midpoint is half way along the line, so it is the average of the coordinates; it is not half the distance.

Worked examples

The treasure: from (−2, 1) to (4, 9). How far in a straight line?

  1. Across: 4 − (−2) = 6. Up: 9 − 1 = 8.
  2. d² = 6² + 8² = 36 + 64 = 100.
  3. d = √100 = 10 m, shorter than the 14 m along the grid lines.

Find the midpoint of (−6, 3) and (2, −5).

  1. x: (−6 + 2) ÷ 2 = −4 ÷ 2 = −2.
  2. y: (3 + (−5)) ÷ 2 = −2 ÷ 2 = −1.
  3. The midpoint is (−2, −1).

Your go. The first step is given. How far apart are P(−3, −1) and Q(5, 4), to 1 decimal place?

  1. Across: 5 − (−3) = 8.
  2. Find the step up, square and add, take the root, round. Then check.
Check the answer

Up: 4 − (−1) = 5. d² = 64 + 25 = 89, and √89 = 9.4 to 1 decimal place.

With a calculator

Find the two differences first, even when a coordinate is negative: for −3 and 5 the difference is 5 − (−3) = 8. Square both, add, then use the square root key on the whole total. Round only at the end, to the accuracy the question asks for.

Check before you go on

1. Which calculation gives the distance between (1, 2) and (4, 6)?

Your turn

Start at Level 1 and go as far as you can. Type negative numbers with a minus sign, for example -7. A wrong answer gets a reason, not just a cross: read it, then try again.

Level 1 · Getting started

2. How far is it from (0, 0) to (21, 28)?

3. Find the x-coordinate of the midpoint of A(−4, 3) and B(10, 7).

Level 2 · Building

4. How far apart are (−5, −2) and (3, 13)?

5. How far apart are (2, −3) and (7, 4)? Give the answer to 1 decimal place.

6. Find the y-coordinate of the midpoint of (−7, −2) and (5, 10).

Level 3 · Stretch

7. The midpoint of AB is (3, −1) and A is (−1, 5). Find the x-coordinate of B.

8. The distance from (0, 0) to (k, 24) is 25, and k is positive. What is k?

Something to think about

Point A is at (2, 5) and B is at (2, 11), directly above it. What does the distance formula give? Does it need the right-angled triangle, or is there a quicker way to see the answer?

One way to think about it

The across step is 0, so d = √(0 + 36) = 6. The triangle has flattened to a line, and the quicker way is to subtract the y-coordinates: 11 − 5 = 6. The formula still works, and it agrees.

Exit ticket

9. Which point is the midpoint of (−3, 5) and (9, −1)?