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MathidiaMathematics Academy
Pythagoras and right triangles
Lesson 59 of 75
Challenge

Pythagoras in context

Objective. Model diagonal-distance problems with right triangles.

  1. 1. Learn
  2. 2. Worked example
  3. 3. Practice
  4. 4. Feedback
  5. 5. Continue

Learn: the key idea

Pythagoras' theorem can model real situations where a right angle is hidden within a diagram, such as a ladder against a wall or the diagonal distance across a rectangular field. First identify the right-angled triangle within the context, labelling the two perpendicular distances and the diagonal line to be found. Then apply a² + b² = c² exactly as usual, remembering the diagonal is always the hypotenuse. Example: a ladder reaching 8 m up a wall with its foot 6 m from the wall has length √(8² + 6²) = √100 = 10 m.

Worked example

A rectangular field is 24 m long and 7 m wide. Find the length of the diagonal path across the field.

Diagram of a rectangle
  1. 1The diagonal forms the hypotenuse of a right-angled triangle with sides 24 m and 7 m.
  2. 2Square both sides: 24² = 576 and 7² = 49.
  3. 3Add the squared values: 576 + 49 = 625.
  4. 4Take the square root: √625 = 25 m.

Answer: 25 m

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Common mistake. Pupils sometimes treat the given diagonal distance as a shorter side instead of correctly identifying it as the hypotenuse; always draw the right-angled triangle first and label the longest side as the hypotenuse.

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