Chapter 1 of 8

All 8 chapters
  1. 1Pythagoras' theorem, readunderstood
  2. 2Using Pythagoras, readunderstood
  3. 3Opposite, adjacent and hypotenuse, readunderstood
  4. 4Sine, cosine and tangent, readunderstood
  5. 5Finding angles, readunderstood
  6. 6Exact values, readunderstood
  7. 7The sine rule and the cosine rule, readunderstood
  8. 8Area of any triangle, and bearings, readunderstood

M4.3.1 Chapter 1

Pythagoras' theorem

A triangle has a right angle and two short sides of 3 cm and 4 cm. How long is the third side?

Your guess

Pick one before you read on. It is fine to be wrong.

Your guess

  • You picked A. Read on. At the end, the chapter comes back to your guess.

  • You picked B. Read on. At the end, the chapter comes back to your guess.

  • You picked C. Read on. At the end, the chapter comes back to your guess.

By the end you can

  • Find the longest side of a right angled triangle
  • Find a shorter side when you know the other two
  • Show that sides 5, 12 and 13 make a right angle

A right angled triangle has one angle of 90°. The side opposite that angle is the longest side. It is called the hypotenuse. The two shorter sides, which meet at the right angle, are the legs.

A right angled triangle ABC with a small square at C. Side AC is labelled leg, side BC is labelled leg, and the long slanted side AB is labelled hypotenuse. The corners are named A, B and C.ABClegleghypotenuse
The hypotenuse faces the right angle. It is always the longest side.

Draw a square on each side of a right angled triangle. Then compare the areas. Ana does this with a triangle whose legs are 4 cm and 3 cm.

Draw it first

Squares on the sides

  1. A right angled triangle with legs 4 cm and 3 cm and a hypotenuse of 5 cm. Step 1: the triangle with the labels 3 cm, 4 cm and 5 cm, and a square on the flat leg with the area 16 cm². Step 2: a square on the upright leg, with the area 9 cm². Step 3: a square on the hypotenuse, with the area 25 cm².4 cm3 cm5 cm16 cm²

    1 Draw a right angled triangle. Its legs are 4 cm and 3 cm. Draw a square on the 4 cm leg. Its area is 4 × 4 = 16 cm².

  2. A right angled triangle with legs 4 cm and 3 cm and a hypotenuse of 5 cm. Step 1: the triangle with the labels 3 cm, 4 cm and 5 cm, and a square on the flat leg with the area 16 cm². Step 2: a square on the upright leg, with the area 9 cm². Step 3: a square on the hypotenuse, with the area 25 cm².4 cm3 cm5 cm16 cm²3 cm9 cm²

    2 Draw a square on the 3 cm leg. Its area is 3 × 3 = 9 cm².

  3. A right angled triangle with legs 4 cm and 3 cm and a hypotenuse of 5 cm. Step 1: the triangle with the labels 3 cm, 4 cm and 5 cm, and a square on the flat leg with the area 16 cm². Step 2: a square on the upright leg, with the area 9 cm². Step 3: a square on the hypotenuse, with the area 25 cm².4 cm3 cm5 cm16 cm²3 cm9 cm²5 cm25 cm²

    3 Draw a square on the hypotenuse. Measure the hypotenuse: it is 5 cm. The square has area 5 × 5 = 25 cm².

Look at the three areas: 16 + 9 = 25. The two small squares have the same total area as the big square. This is true for every right angled triangle. It is called Pythagoras' theorem.

Why is it true? Draw two squares with the same side, a + b. Each holds four copies of the triangle, all with legs a and b. Take the four triangles away from each square.

Two squares of the same size, each with side a + b. In the left square, four hatched triangles surround a square a² and a square b². In the right square, the same four hatched triangles surround one tilted square c². The top edges are marked a and b, and each square is labelled a + b.a²b²c²aba + baba + b
Same big square, same four triangles. So the white parts have equal area: a² + b² = c².

The big squares are equal. The four triangles are equal. So what is left must be equal too: a² + b² on the left, c² on the right. That is a proof, not a guess.

In a right angled triangle, a² + b² = c². Here c is the hypotenuse. The rule works only when the triangle has a right angle.

To find the hypotenuse, add the squares of the legs, then take the square root. We take the positive root, because a length is above 0.

A right angled triangle with legs 6 cm and 8 cm and the hypotenuse labelled c. The corners are named A, B and C.ABC6 cm8 cmc

Worked example

A right angled triangle has legs of 6 cm and 8 cm. Find the hypotenuse.

  1. The legs are a = 6 and b = 8. We want the hypotenuse c.
  2. Use c² = a² + b² = 6² + 8².
  3. Square each leg and add: 36 + 64 = 100.
  4. Take the square root: c = √100 = 10.

Answer: The hypotenuse is 10 cm.

The answer is not always a whole number. When it is not, the calculator gives a long decimal. Say how you rounded.

A right angled triangle with legs 5 cm and 7 cm and the hypotenuse labelled c. The corners are named A, B and C.ABC5 cm7 cmc

Worked example

A right angled triangle has legs of 5 cm and 7 cm. Find the hypotenuse to 1 decimal place.

  1. Use c² = 5² + 7².
  2. Square each leg: 5² = 25 and 7² = 49.
  3. Add: c² = 25 + 49 = 74.
  4. Take the square root: c = √74 = 8.602 on the calculator. To 1 decimal place that is 8.6.

Answer: To 1 decimal place, the hypotenuse is 8.6 cm.

Now the hypotenuse is known and a leg is not. Rewrite the rule. If c² = a² + b², then a² = c² − b². So you subtract. The hypotenuse is always the one you start from.

A right angled triangle with the hypotenuse 25 cm, one leg 7 cm and the other leg labelled a. The corners are named A, B and C.ABCa7 cm25 cm

Worked example

A right angled triangle has hypotenuse 25 cm and one leg 7 cm. Find the other leg.

  1. The hypotenuse is c = 25. One leg is b = 7. We want the other leg a.
  2. Use a² = c² − b² = 25² − 7².
  3. Square each number: 25² = 625 and 7² = 49.
  4. Subtract: a² = 625 − 49 = 576.
  5. Take the square root: a = √576 = 24.

Answer: The other leg is 24 cm.

The rule also works backwards. Suppose a triangle has sides 5 cm, 12 cm and 13 cm. Is there a right angle? Try the sum with c as the longest side.

A triangle with sides 5 cm, 12 cm and 13 cm. The corners are named A, B and C. No corner is marked as a right angle yet.ABC5 cm12 cm13 cm
Is the corner between the 5 cm and 12 cm sides a right angle? The sums will tell us.

Worked example

Do sides of 5 cm, 12 cm and 13 cm make a right angled triangle?

  1. The longest side is 13. If there is a right angle, it is opposite this side, so c = 13.
  2. The squares of the two shorter sides are 5² = 25 and 12² = 144.
  3. Add them: 25 + 144 = 169.
  4. The square of the longest side is 13² = 169.
  5. The two results are equal, so a² + b² = c² holds. The triangle has a right angle.

Answer: Yes. 25 + 144 = 169, and 13 × 13 = 169, so the triangle has a right angle.

The same triangle with sides 5 cm, 12 cm and 13 cm, now with a small square at the corner between the two shorter sides. The corners are named A, B and C.ABC5 cm12 cm13 cm
The sums matched, so the corner opposite the 13 cm side is a right angle. The longest side faces it.

Why can we trust it? A triangle is fixed by its three sides. Build the right angled triangle with legs 5 and 12. Its hypotenuse is 13. Any triangle with sides 5, 12 and 13 is the same triangle, so it has a right angle too. The sums must match exactly. Sides 4, 5 and 6 give 16 + 25 = 41, but 6 × 6 = 36. So they make no right angle.

See it three ways

With things

A builder ties a loop of string with 12 equal spaces. Pulled tight with 3 spaces, 4 spaces and 5 spaces, it makes a square corner.

As a picture

A right angled triangle with sides 3 spaces, 4 spaces and 5 spaces and a small square at the right angle. The corners are named A, B and C.ABC3 spaces4 spaces5 spaces

In symbols

3² + 4² = 9 + 16 = 25 = 5².

Try it

A square corner from string

You need: a piece of string about 60 cm long, a pencil and a ruler, a book with a square corner.

  1. Mark the string every 5 cm, so there are 12 equal spaces. Tie the ends together.
  2. Hold the loop at three marks so the sides are 3 spaces, 4 spaces and 5 spaces long. Pull it tight.
  3. Put the corner of the book into the corner between the 3 and the 4. Compare.

What you will see: The corner of the string fits the corner of the book. 3, 4 and 5 make a right angle, because 3² + 4² = 5².

If you cannot try it, this is what happens.
A right angled triangle of string with sides 3 spaces, 4 spaces and 5 spaces and a small square at the right angle. The corners are named A, B and C.ABC3 spaces4 spaces5 spaces

A wrong idea many people have

a² + b² = c² works for every triangle.

Why it feels right: The first triangles we meet in this chapter all have right angles. So the rule seems to belong to every triangle.

The test that breaks it: Take a triangle with sides 4, 5 and 7. Then 4² + 5² = 16 + 25 = 41, but 7² = 49. The two results are not equal.

What is true: The rule holds only when the triangle has a right angle, and c is the side opposite it. When the sums do not match, there is no right angle.

A wrong idea many people have

The hypotenuse is a + b, because it joins the ends of the two legs.

Why it feels right: Walking along both legs and walking across the corner look like the same trip.

The test that breaks it: In the 3, 4, 5 triangle, a + b = 3 + 4 = 7, but the hypotenuse is 5 cm. The short cut across is shorter than the way round.

What is true: The hypotenuse is less than a + b. Square first, add, then take the root: c = √(a² + b²).

Key words

hypotenuse
the longest side of a right angled triangle, opposite the right angle.
Pythagoras' theorem
in a right angled triangle, a² + b² = c², with c the hypotenuse.

Back to your guess

The answer is B: 5 cm, because 3 × 3 + 4 × 4 = 5 × 5.

The third side is 5 cm. It cannot be 7 cm: in a triangle, one side is always shorter than the other two added together. The sum 3 × 3 + 4 × 4 = 25 = 5 × 5 is Pythagoras' theorem, and this chapter shows why it is true.

Check yourself

Try each one before you look. A wrong answer tells you where to go next.

  1. Which side is the hypotenuse of this triangle?

    A right angled triangle PQR with a small square at Q. The corners are named P, Q and R. The side PQ is short, the side QR is short, and the side PR is long.PQR
    Tap your answer.

    1. Not this one. PQ is a leg: it meets the right angle at Q. The hypotenuse does not touch the right angle. Go to M4.3.1. Try another.

    2. Yes, that is right. The right angle is at Q. The side opposite it is PR, so PR is the hypotenuse. It is also the longest side.

    3. Not this one. QR is a leg too. It ends at the right angle. The hypotenuse is the side facing the right angle. Go to M4.3.1. Try another.

    4. Not this one. It cannot. The hypotenuse is always the side opposite the right angle, and it is always the longest side. Go to M4.3.1. Try another.

    See the answer

    PR

    The right angle is at Q. The side opposite it is PR, so PR is the hypotenuse. It is also the longest side.

    • If you chose A: PQ is a leg: it meets the right angle at Q. The hypotenuse does not touch the right angle. Go to M4.3.1.
    • If you chose C: QR is a leg too. It ends at the right angle. The hypotenuse is the side facing the right angle. Go to M4.3.1.
    • If you chose D: It cannot. The hypotenuse is always the side opposite the right angle, and it is always the longest side. Go to M4.3.1.

  2. The legs of a right angled triangle are 9 cm and 12 cm. How long is the hypotenuse?

    A right angled triangle with legs 9 cm and 12 cm and the hypotenuse labelled c. The corners are named A, B and C.ABC9 cm12 cmc
    Tap your answer.

    1. Not this one. You added the legs, 9 + 12. The hypotenuse is shorter than the two legs added together. Square first, add, then take the root. Go to M4.3.1. Try another.

    2. Not this one. 225 is c², the square of the hypotenuse: 81 + 144. Take the square root to find c itself. Go to M4.3.1. Try another.

    3. Not this one. You took the shorter leg from the longer one. For a hypotenuse you add the squares, not subtract the lengths. Go to M4.3.1. Try another.

    4. Yes, that is right. c² = 9² + 12² = 81 + 144 = 225, and √225 = 15.

    See the answer

    15 cm

    c² = 9² + 12² = 81 + 144 = 225, and √225 = 15.

    • If you chose A: You added the legs, 9 + 12. The hypotenuse is shorter than the two legs added together. Square first, add, then take the root. Go to M4.3.1.
    • If you chose B: 225 is c², the square of the hypotenuse: 81 + 144. Take the square root to find c itself. Go to M4.3.1.
    • If you chose C: You took the shorter leg from the longer one. For a hypotenuse you add the squares, not subtract the lengths. Go to M4.3.1.

  3. The hypotenuse of a right angled triangle is 17 cm and one leg is 8 cm. How long is the other leg?

    A right angled triangle with the hypotenuse 17 cm, one leg 8 cm and the other leg labelled a. The corners are named A, B and C.ABCa8 cm17 cm
    Tap your answer.

    1. Yes, that is right. a² = 17² − 8² = 289 − 64 = 225, and √225 = 15.

    2. Not this one. You added the squares, as for a hypotenuse. Here the hypotenuse is given, so subtract: 17² − 8². Go to M4.3.1. Try another.

    3. Not this one. You took 8 from 17. Subtract the squares, 289 − 64, and then take the square root. Go to M4.3.1. Try another.

    4. Not this one. You added 17 and 8. A leg is shorter than the hypotenuse, and 25 is longer than 17. Go to M4.3.1. Try another.

    See the answer

    15 cm

    a² = 17² − 8² = 289 − 64 = 225, and √225 = 15.

    • If you chose B: You added the squares, as for a hypotenuse. Here the hypotenuse is given, so subtract: 17² − 8². Go to M4.3.1.
    • If you chose C: You took 8 from 17. Subtract the squares, 289 − 64, and then take the square root. Go to M4.3.1.
    • If you chose D: You added 17 and 8. A leg is shorter than the hypotenuse, and 25 is longer than 17. Go to M4.3.1.

  4. Which set of sides, in cm, makes a right angled triangle?

    Tap your answer.

    1. Not this one. Test it: 5² + 6² = 25 + 36 = 61, but 8² = 64. They are not equal, so there is no right angle. Go to M4.3.1. Try another.

    2. Not this one. Test it: 4² + 5² = 16 + 25 = 41, but 6² = 36. They are not equal. Close numbers are not enough. Go to M4.3.1. Try another.

    3. Yes, that is right. 8² + 15² = 64 + 225 = 289 and 17² = 289. The sums are equal, so the triangle has a right angle. In each other set the sums differ.

    4. Not this one. Test it: 6² + 7² = 36 + 49 = 85, but 10² = 100. They are not equal, so there is no right angle. Go to M4.3.1. Try another.

    See the answer

    8, 15, 17

    8² + 15² = 64 + 225 = 289 and 17² = 289. The sums are equal, so the triangle has a right angle. In each other set the sums differ.

    • If you chose A: Test it: 5² + 6² = 25 + 36 = 61, but 8² = 64. They are not equal, so there is no right angle. Go to M4.3.1.
    • If you chose B: Test it: 4² + 5² = 16 + 25 = 41, but 6² = 36. They are not equal. Close numbers are not enough. Go to M4.3.1.
    • If you chose D: Test it: 6² + 7² = 36 + 49 = 85, but 10² = 100. They are not equal, so there is no right angle. Go to M4.3.1.

  5. The squares on the two legs of a right angled triangle have areas 36 cm² and 64 cm². What is the area of the square on the hypotenuse?

    A right angled triangle with a square on each side. The squares on the legs are labelled a² and b², the square on the hypotenuse is labelled c², and the sides are labelled a, b and c.abca²b²c²
    Tap your answer.

    1. Not this one. Ten is the hypotenuse itself, √(36 + 64). The question asks for the area of the square on it, which is 100. Go to M4.3.1. Try another.

    2. Yes, that is right. The two small squares have the same total area as the big one: 36 + 64 = 100 cm².

    3. Not this one. You added the side lengths, 6 and 8. The areas are 36 and 64, and those are what you add. Go to M4.3.1. Try another.

    See the answer

    100 cm²

    The two small squares have the same total area as the big one: 36 + 64 = 100 cm².

    • If you chose A: Ten is the hypotenuse itself, √(36 + 64). The question asks for the area of the square on it, which is 100. Go to M4.3.1.
    • If you chose C: You added the side lengths, 6 and 8. The areas are 36 and 64, and those are what you add. Go to M4.3.1.

  6. Which triangle can you use a² + b² = c² on?

    Tap your answer.

    1. Yes, that is right. The proof used a right angle: four triangles with a square corner fit together into squares. Without a right angle the proof, and the rule, fail.

    2. Not this one. It fails for most triangles. For sides 4, 5 and 7, 4² + 5² = 41 but 7² = 49. The rule needs a right angle. Go to M4.3.1. Try another.

    3. Not this one. Whole sides are not needed. A right angled triangle with legs 1 cm and 1 cm has a hypotenuse of √2 cm, about 1.41 cm. The rule still holds. Go to M4.3.1. Try another.

    See the answer

    A triangle with a right angle, where c is the side opposite it.

    The proof used a right angle: four triangles with a square corner fit together into squares. Without a right angle the proof, and the rule, fail.

    • If you chose B: It fails for most triangles. For sides 4, 5 and 7, 4² + 5² = 41 but 7² = 49. The rule needs a right angle. Go to M4.3.1.
    • If you chose C: Whole sides are not needed. A right angled triangle with legs 1 cm and 1 cm has a hypotenuse of √2 cm, about 1.41 cm. The rule still holds. Go to M4.3.1.

  7. The legs of a right angled triangle are 20 cm and 21 cm. How many centimetres is the hypotenuse?

    See the answer

    29

    c² = 20² + 21² = 400 + 441 = 841, and √841 = 29.

  8. The hypotenuse of a right angled triangle is 26 cm and one leg is 10 cm. How many centimetres is the other leg?

    See the answer

    24

    a² = 26² − 10² = 676 − 100 = 576, and √576 = 24.

  9. The legs of a right angled triangle are 4 cm and 6 cm. How long is the hypotenuse, to 1 decimal place?

    See the answer

    7.2

    c² = 4² + 6² = 16 + 36 = 52. The calculator gives √52 = 7.211, which is 7.2 to 1 decimal place.

Keep it

Copy this into your exercise book.

In a right angled triangle, a² + b² = c². The hypotenuse c faces the right angle. Add squares to find c. Subtract squares to find a leg.

Draw: A right angled triangle with a square on each side, labelled a², b² and c², and a² + b² = c² under it.

Reading with a child? A short page for the adult: what this chapter teaches and three questions to ask.

No internet tomorrow? Save the whole book as one file (776 KB). It opens on any phone.