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      Lesson details

      Learning outcome

      I can identify which side length is missing and use the appropriate form of Pythagoras' theorem to calculate it.

      Key learning points

      1. The hypotenuse is the side opposite the right angle.
      2. It does not matter which of the shorter sides is identified as a or b.

      Keywords

      • Right-angled triangle - A right-angled triangle has exactly one 90° interior angle.

      • Hypotenuse - A hypotenuse is the side of the right-angle triangle which is opposite the right-angle.

      • 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

      I can only use Pythagoras' theorem to find the length of a side of a right-angled triangle when the right-angle marker is labelled, so I know which side is the hypotenuse.

      The right-angle marker doesn't need to be explicitly labelled in order to be able to identify the right-angle. You can use knowledge of "interior angles in a triangle sum to 180°" to find the right angle, given two other angles.

      Teacher tip

      It is important to acknowledge the commutativity of the squares of the two shorter sides when using Pythagoras' theorem; there is more than one way of presenting a Pythagoras' theorem equation, and different arrangements may be more beneficial in different scenarios.

      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.
      In this diagram, $$a$$ cm² = 140 cm² and $$b$$ cm² = 185 cm². The area $$c$$ cm² is cm².

      An image in a quiz
      Correct Answer: 325, 325 cm², 325cm², 325 cm squared, 325cm squared

      Q2.
      In this diagram, $$b$$ cm² = 77 cm² and $$c$$ cm² = 107 cm². Find the area $$a$$ cm².

      An image in a quiz
      5.48 cm²
      13.56 cm²
      Correct answer: 30 cm²
      184 cm²
      4119.5 cm²

      Q3.
      In the diagram, $$a$$ = 1504 and $$b$$ = 1200. Find the length of the hypotenuse of the triangle.

      An image in a quiz
      17.44 cm
      Correct answer: 52 cm
      304 cm
      1924.06 cm
      2704 cm

      Q4.
      In this diagram, $$a$$ = 49, $$b$$ = 576, and $$c$$ = 625. The perimeter of the triangle is cm.

      An image in a quiz
      Correct Answer: 56, 56cm, 56 cm

      Q5.
      In a right-angled triangle, one of its acute interior angles is 33°. What is the size of its other acute interior angle?

      33°
      Correct answer: 57°
      90°
      147°
      impossible to tell

      Q6.
      The size of an acute angle in a right-angled isosceles triangle is °.

      Correct Answer: 45, 45°, 45 degrees, 45degrees

      6 Questions

      Q1.
      Which of these triangles are right-angled triangles?

      An image in a quiz
      a
      Correct answer: b
      Correct answer: c
      d
      Correct answer: e

      Q2.
      In the diagram, $$x$$ = 101, $$y$$ = 45, $$z$$ = 56. Which of these angles is the right angle?

      An image in a quiz
      $$a$$°
      Correct answer: $$b$$°
      $$c$$°
      impossible to tell, but this is a right-angled triangle
      this is not a right-angled triangle

      Q3.
      A right-angled triangle has sides of length 21 inches, 220 inches, and 221 inches. Which of these are correct Pythagoras' theorem equations for this triangle?

      21 + 220 = 221
      Correct answer: 21² + 220² = 221²
      Correct answer: 221² = 220² + 21²
      220² = 221² + 21²
      221² + 21² = 220²

      Q4.
      A right-angled triangle has sides of length 39 inches, 80 inches, and 89 inches. Which of these are correctly rearranged Pythagoras' theorem equations for this triangle?

      Correct answer: 89² – 39² = 80²
      89² + 39² = 80²
      Correct answer: 89² – 80² = 39²
      89² = 80² – 39²
      Correct answer: 0 = 89² – 80² – 39²

      Q5.
      Using Pythagoras' theorem, calculate the length of the side labelled $$j$$, rounded to 1 decimal place. Length of side $$j$$ is cm.

      An image in a quiz
      Correct Answer: 22.6, 22.6 cm, 22.6cm

      Q6.
      Using Pythagoras' theorem twice, calculate the length of the side labelled $$h$$, rounded to 2 decimal places. Length of side $$h$$ is cm.

      An image in a quiz
      Correct Answer: 20.69, 20.69 cm, 20.69cm

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