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      Demonstrating Pythagoras' theorem

      Lesson details

      Learning outcome

      I can appreciate there is a relationship between the lengths of the sides of a right-angled triangle.

      Key learning points

      1. A visual approach can help you understand the structure behind Pythagoras' theorem.
      2. There is a difference between proof and demonstration.
      3. A demonstration would be showing Pythagoras' theorem works for specific right-angled triangles.
      4. A proof is generalised i.e. using four congruent triangles arranged in a particular way inside a square.
      5. The sum of the squares of the two shorter sides equals the square of the longest side.

      Keywords

      • Pythagoras' theorem - Pythagoras’ theorem shows that the sum of the squares of the two shorter sides of a right-angled triangle is equal to the square of its longest side (the hypotenuse).

      Common misconception

      Pythagoras' theorem is just a relationship between the three sides of a right-angled triangle.

      Whilst this is true, Pythagoras' theorem can more visually be represented as three squares whose sides are equal in length to the three sides of the triangle. The sum of the areas of the two smaller squares is equal to the area of the larger square.

      Teacher tip

      When students are identifying whether the largest angle in a triangle they have constructed, the angle may be ambiguous. Advise them to use a protractor with caution, as accurate measuring of the angles may be tricky with several moving pieces.

      Licence

      This content is © Oak National Academy Limited (2025), licensed on Open Government Licence version 3.0
      except where otherwise stated. See Oak's terms & conditions
      (Collection 2).

      Lesson video

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      Prior knowledge starter quiz

      6 Questions

      Q1.
      Which of these are square numbers?

      Correct answer: 0
      Correct answer: 1
      2
      Correct answer: 4
      8

      Q2.
      Match each statement to its value.

      Correct Answer:5 squared,25
      Correct Answer:the square of 4,16
      Correct Answer:12²,144
      Correct Answer:20 × 20,400
      Correct Answer:$$\sqrt{4}$$,2
      Correct Answer:$$\sqrt{25}$$,5

      Q3.
      The difference between 11² and 7² is .

      Correct Answer: 72

      Q4.
      8² + 6² – 10² = .

      Correct Answer: 0

      Q5.
      Which of these show a fully and correctly correctly marked square?

      An image in a quiz
      shape A
      shape B
      shape C
      Correct answer: shape D
      shape E

      Q6.
      Starting with the smallest, place these angles in order of size.

      1 -
      2 - acute angle
      3 - right angle
      4 - obtuse angle
      5 - reflex angle
      6 - angle around a point that makes one full turn

      6 Questions

      Q1.
      What is the size of the largest angle in the triangle formed from these three squares?

      An image in a quiz
      acute
      Correct answer: obtuse
      reflex
      right
      impossible to tell

      Q2.
      Which of these three angles is the largest?

      An image in a quiz
      $$x$$°
      Correct answer: $$y$$°
      $$z$$°
      impossible to tell

      Q3.
      Which of these are possible sizes for the largest angle?

      An image in a quiz
      46°
      Correct answer: 68°
      90°
      124°
      173°

      Q4.
      If three congruent squares are joined at their vertices, what type of triangle is formed?

      Correct answer: equilateral
      isosceles
      isosceles, right-angled
      scalene
      scalene, right-angled

      Q5.
      Two congruent squares, A and B, and a third square, C, are joined at their vertices. The area of square C is less than the area of square A. What type of triangle is formed?

      equilateral
      Correct answer: isosceles
      isosceles, right-angled
      scalene
      scalene, right-angled

      Q6.
      A right-angled triangle is formed from three squares. The area of two of the squares are 50 units² and 70 units². What are the possible areas of the third square?

      $$7\over5$$ units²
      Correct answer: 20 units²
      65 units²
      Correct answer: 120 units²
      It is impossible for a right-angled triangle to be made from these squares.

      To help you plan your 9 maths lesson on: Demonstrating Pythagoras' theorem, download all teaching resources for free and adapt to suit your pupils' needs...