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SavannahYear 8
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MathsNumber, algebra and geometry: connected foundations32 min★★★ difficulty

Chapter 9: Pythagoras and geometric proof

Pythagoras links the three sides of every right-angled triangle and can be justified using areas of squares.

Part of your national curriculum
  • Geometry and measures: Use Pythagoras' theorem to solve problems involving right-angled triangles

Lesson overview

What you'll learn in this lesson

Use Pythagoras' theorem to solve problems involving right-angled triangles

Key learning points

  • • The relationship
  • • Find a side
  • • Problem solving

This lesson at a glance

  • 32 minutes
  • 14 parts to scroll through
  • 3 quick checks
  • Marked quiz at the end
  • Gentle pace: short sittings with pauses

Words to know

pythagorastheoremproblemsinvolvingright-angled

Scroll down — the lesson carries on below

Part 1 of 14 · Discover7%
1

Watch & discover

Part 1 of 14

Watch first

Chapter 9: Pythagoras and geometric proof

An English teaching video on this topic from mathantics. Watch it, then scroll down for the lesson.

In a nutshell

Use Pythagoras' theorem to solve problems involving right-angled triangles

2

Learning cycle

Part 2 of 14

Learning cycle 1 of 2

Part 1 · The relationship

A short piece of teaching, then a check to make sure it has landed.

3

Explore the idea

Part 3 of 14

Learn

The relationship

For a right-angled triangle, a²+b²=c², where c is the hypotenuse opposite the right angle. The theorem does not apply directly to non-right-angled triangles.

4

Explore the idea

Part 4 of 14

Learn

Find a side

With shorter sides 6 cm and 8 cm, c=√(36+64)=10 cm. If c=13 cm and one shorter side is 5 cm, the other is √(169−25)=12 cm.

5

Quick check

Part 5 of 14

Quick check

A right triangle has shorter sides 6 and 8. Find c.

6

Reset break

Part 6 of 14

Pause

That's sitting 1 of 2 done

Stretch, get a drink, look out of the window. There is no timer and nothing is counting down — your place is saved, so you can come back in five minutes or tomorrow.

Stop here for now
7

Quick check

Part 7 of 14

Quick check

The hypotenuse is 13 and one side is 5. Find the other.

8

Learning cycle

Part 8 of 14

Learning cycle 2 of 2

Part 2 · Problem solving

A short piece of teaching, then a check to make sure it has landed.

9

Explore the idea

Part 9 of 14

Learn

Problem solving

Sketch and label the right triangle, identify the hypotenuse, substitute with units, calculate, then round only at the end. Compare the answer with the longest-side condition.

10

Quick check

Part 10 of 14

Quick check

Which side is c?

11

Challenge round

Part 11 of 14

Game · Sort it

Which of these are true?

Drag each card into the right column. Tap a card first if dragging is fiddly.

True

Not true

12

Challenge round

Part 12 of 14

Game · Fill the gaps

Finish the sentences

Choose the word that belongs in each gap.

For a ____ triangle, a²+b²=c², where c is the hypotenuse opposite the right angle.

The ____ does not apply directly to non-right-angled triangles.

13

Mastery quiz

Part 13 of 14

Marked quiz

End of lesson quiz: Chapter 9: Pythagoras and geometric proof

3 questions, marked with the reasoning shown. No timer.

  1. 1. A right triangle has shorter sides 6 and 8. Find c.

  2. 2. The hypotenuse is 13 and one side is 5. Find the other.

  3. 3. Which side is c?

14

Lesson round-up

Part 14 of 14

Lesson round-up

Ready when you are

Quiz score

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Games

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Points this lesson

0

Best run

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Luna: 0 out of 3 on the practice checks. Only if you feel up to it — one more?

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Do the work

Your written work for this lesson

This is the part you'd do in your book at school. Pick the pieces below, write here or on paper, and photograph anything you draw or work out.

Article

The cost of a school trip

A school is planning a trip to a wildlife park. The coach costs £480 however many people go. Entry costs £9.50 for each pupil and each adult goes free. The school says every pupil must pay the same amount, and the trip must not lose money.

If 60 pupils go, the coach works out at £8 each, so the total for each pupil is £17.50. If only 40 pupils go, the coach works out at £12 each, so the price rises to £21.50. The teacher explains that the coach is a fixed cost and the entry is a cost per person — which is why the price per pupil changes when the number changes.

The school decides to charge £20 and to use anything left over for a second trip later in the year.

Written for Good Learning Co.

Reading questions

  1. Work out the total cost of the trip if 50 pupils go.
  2. Show why the price per pupil falls when more pupils go, using numbers.
  3. At £20 each, how many pupils must go before the trip makes money? Show your working.

  • Working out

    Working out

    Do the practice questions on paper, showing every step — not just the answer. Mark them yourself and circle anything you got wrong.

    10–15 questions

  • Worked example, explained

    Writing

    Copy one question out and write underneath, in words, why each step works.

    1 question, 4–5 sentences

  • Problem in words

    Working out

    Write your own real-life problem that uses today's method, then solve it.

    1 problem + solution

Saved as you type.

Part 1 of 14 · Watch & discover

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    Use Pythagoras' theorem to solve problems involving right-angled triangles

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