Year 10 Standard · Unit 2 Generalizing and coordinate geometry · Lesson 3 of 6

How far apart, and where is halfway?

Two points on a grid make the corners of a right-angled triangle, so Pythagoras gives their distance. The halfway point is found by averaging each coordinate. Both are general rules for any two points.

Number and AlgebraGDC optional

Warm-up

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

  1. From Year 9: a right-angled triangle has legs 3 cm and 4 cm. How long is the hypotenuse?
    Answer5 cm, because 9 + 16 = 25.
  2. From Year 8: what is the number halfway between −6 and 10?
    Answer2, because (−6 + 10) ÷ 2 = 2.
  3. From Year 9: what is the length of the horizontal segment from (−3, 4) to (5, 4)?
    Answer8, because 5 − (−3) = 8.
Words used in this lesson
distance
the length of the straight line between two points. For example: AB.
midpoint
the point exactly halfway along a line segment. For example: M.
coordinates
a pair of numbers that give a point's position. For example: (2, 5).
hypotenuse
the longest side of a right-angled triangle. For example: the line AB.
segment
the part of a line between two points. For example: PQ.
average
the sum divided by the number of values. For example: (a + b) ÷ 2.

Across and up, or straight?

On a map of a nature reserve in Kenya, one unit is 1 km. A ranger plans to walk from gate G at (1, 2) to gate H at (7, 10) along the paths of the grid: 6 km across and 8 km up, 14 km in all. A drone will fly the straight line between the gates.

Predict first: will the straight line be shorter than 14 km, longer, or the same? Write down a number if you can. Then use the figure: set A and B and see the triangle between them.

−3
−1
2
3
√41 ≈ 6.40distance AB
(−0.5, 1)midpoint M

A(−3, −1) and B(2, 3): across 5, up 4, so AB² = 5² + 4² = 41 and AB = √41 ≈ 6.40. The midpoint is (−0.5, 1).

Show me the steps
  1. A(−3, −1) and B(2, 3): across 5, up 4. AB² = 25 + 16 = 41, so AB = √41 ≈ 6.40. The midpoint is (−0.5, 1).
  2. A(0, 0) and B(3, 4): across 3, up 4. AB² = 9 + 16 = 25, so AB = 5. The midpoint is (1.5, 2).
  3. A(−4, 2) and B(2, 2): same height, so up 0 and AB = 6 across. The midpoint is (−1, 2).
  4. A(−5, 1) and B(4, −3): across 9, up −4 (down 4). AB² = 81 + 16 = 97. The sign of a difference does not matter once it is squared.

Distance: for A(x₁, y₁) and B(x₂, y₂), AB = √((x₂ − x₁)² + (y₂ − y₁)²). The across and up differences are the legs of a right-angled triangle, and the distance is its hypotenuse. Squaring means the order of subtraction does not matter.

Midpoint: M = ((x₁ + x₂) ÷ 2, (y₁ + y₂) ÷ 2), the average of each coordinate. To find a missing end point B from A and the midpoint M, go from A to M and then the same again: B = 2M − A, coordinate by coordinate.

Two common mistakes. Adding the differences instead of squaring them: across 3 and up 4 make 5, not 7. And forgetting to halve for the midpoint: (3 + 5) is 8, but halfway between 3 and 5 is 4.

Worked examples

1. Find the distance between P(−4, −3) and Q(5, 37).

  1. Across: 5 − (−4) = 9. Up: 37 − (−3) = 40.
  2. PQ² = 9² + 40² = 81 + 1600 = 1681, so PQ = 41.

2. Find the midpoint of (−4, 7) and (6, −1).

  1. x: (−4 + 6) ÷ 2 = 1. y: (7 + (−1)) ÷ 2 = 3.
  2. The midpoint is (1, 3).

3. The midpoint of AB is M(3, 1) and A is (−1, 5). Find B.

  1. From A to M the x changes by 3 − (−1) = 4 and the y changes by 1 − 5 = −4.
  2. Go the same again from M: B = (3 + 4, 1 + (−4)) = (7, −3).

Your go. The first step is given. Find the distance between C(1, −2) and D(4, 7). Give it exactly, as a surd in its simplest form. The across difference is 4 − 1 = 3.

  1. Across: 3. Now find the up difference: 7 − (−2).
  2. Square both and add, then take the square root and simplify.
Check the answer

Up: 7 − (−2) = 9. CD² = 3² + 9² = 9 + 81 = 90, so CD = √90 = 3√10 (about 9.49).

With a GDC (optional)

A GDC can give the decimal value of a distance. Type the square root of the sum, for example √(41), and read the decimal. It is useful for checking that a surd such as √41 is about 6.4. Keep the exact form in your working and round only at the end, to the accuracy the question asks for.

Some GDCs have a distance command, but it is quicker and safer to write the squared differences by hand. Everything here can be done by hand.

Check before you go on

1. What is the midpoint of (2, 8) and (10, 2)?

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. What is the distance from the origin (0, 0) to the point (9, 12)?

3. What is the x-coordinate of the midpoint of (−7, 4) and (3, 10)?

Level 2 · Building

4. What is the distance between P(−3, 2) and Q(5, 17)?

5. What is the y-coordinate of the midpoint of (4, −9) and (−2, 3)?

6. M(2, 5) is the midpoint of AB. A is (−1, 3). What is the y-coordinate of B?

Level 3 · Stretch

7. A cable car in Peru runs from a valley station at (100, 300) to a summit station at (380, 1260). Both are measured in metres on a map grid. What is the straight-line length of the cable, in metres?

8. A is (−1, 3) and B is (11, k) with k greater than 3. The distance AB is 13. What is k?

Something to think about

A point is exactly 25 units from the origin. Its coordinates can both be whole numbers in more than one way. Find at least three such points, and say how you know each is exactly 25 away.

One way to think about it

x² + y² = 625. (25, 0) and (0, 25) work, and so do (7, 24) and (15, 20), because 49 + 576 = 625 and 225 + 400 = 625. Changing signs and swapping x and y gives 20 points in all.

Exit ticket

9. Which statement is correct for the points P(−6, −5) and Q(14, 16)?