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12.1 Pythagoras' Theorem 1

GCSE Maths Foundation · Edexcel · Chapter 12: Right-angled triangles

A free GCSE Maths Foundation lesson for Edexcel (9–1) from OBA School: the hypotenuse, Pythagoras’ theorem, finding the longest side and checking for a right angle. Watch the video, read the explanation and worked examples, then try the revision questions. You can work through the full lesson, with interactive activities, practice quizzes and printable worksheets, in our free online course.

What you will learn

  • Identify the hypotenuse of a right-angled triangle
  • Use Pythagoras’ theorem to find the length of the hypotenuse
  • Use the converse of Pythagoras’ theorem to decide whether a triangle is right-angled
  • Solve problems such as diagonals and distances between points

Key words

  • Hypotenuse: The longest side of a right-angled triangle. It is opposite the right angle.
  • Pythagoras’ theorem: In a right-angled triangle, a² + b² = c², where c is the hypotenuse.
  • Converse: The theorem used the other way round: if a² + b² = c², the triangle has a right angle.
  • Pythagorean triple: Three whole numbers that fit the theorem, such as 3, 4, 5 or 5, 12, 13.

Pythagoras’ theorem

Draw a square on each side of a right-angled triangle. The area of the square on the hypotenuse equals the total area of the other two squares.

Key fact: a² + b² = c². The hypotenuse c is always opposite the right angle and is the longest side.

Worked example: checking the rule

A triangle has a right angle and shorter sides 5 cm and 12 cm. Show that the hypotenuse is 13 cm.
5² + 12² = 25 + 144 = 169. √169 = 13

Answer: 13 cm

Finding the hypotenuse

Square the two shorter sides, add, then square root. Round only at the end. On a calculator you can type √(5² + 9²) in one go.

Worked example: rounding

The shorter sides of a right-angled triangle are 5 cm and 9 cm. Find the hypotenuse to 1 decimal place.
c² = 5² + 9² = 25 + 81 = 106
c = √106 = 10.295… = 10.3 cm

Answer: 10.3 cm

Is the triangle right-angled?

Square the two shorter sides and add. Compare with the square of the longest side. Equal means a right angle. Not equal means no right angle.

  • 7, 24, 25; Shorter sides squared and added 49 + 576 = 625; Longest side squared 625; Right-angled? yes
  • 6, 9, 11; Shorter sides squared and added 36 + 81 = 117; Longest side squared 121; Right-angled? no
  • 8, 15, 17; Shorter sides squared and added 64 + 225 = 289; Longest side squared 289; Right-angled? yes

Worked example: a picture frame

A frame is 30 cm by 40 cm and its diagonal measures 50 cm. Are the corners right angles?
30² + 40² = 900 + 1600 = 2500 and 50² = 2500. They are equal.

Answer: Yes, the corners are right angles

Problems: diagonals and distances

Look for a hidden right-angled triangle. A rectangle’s diagonal is a hypotenuse. The distance between two points on a grid is the hypotenuse of a triangle with a horizontal and a vertical side.

Worked example: distance between points

Find the distance between P(1, 1) and Q(5, 4).
Across: 5 − 1 = 4. Up: 4 − 1 = 3.
PQ² = 4² + 3² = 25, so PQ = 5

Answer: 5 units

Common mistakes

  • Forgetting to square root at the end: the answer is c, not c².
  • Adding the sides before squaring: (6 + 8)² is not 6² + 8².
  • Using the theorem in a triangle that has no right angle.
  • Rounding in the middle of the working. Keep the full value until the end.

Revision questions

Try these, then check your answers below each one.

Question 1. Triangle ABC has a right angle at B. Which side is the hypotenuse? (Choose from: AB, BC, AC.)

Answer: AC

Working: The right angle is at B, so the side opposite it is AC.

Question 2. The two shorter sides of a right-angled triangle are 9 cm and 40 cm. Work out the hypotenuse.

Answer: 41 cm

Working: 9² + 40² = 81 + 1600 = 1681. √1681 = 41 cm.

Question 3. The two shorter sides of a right-angled triangle are 6 cm and 11 cm. Work out the hypotenuse, correct to 1 decimal place.

Answer: 12.5 cm

Working: 6² + 11² = 36 + 121 = 157. √157 = 12.529… ≈ 12.5 cm.

Related lessons

Continue in the full course

This lesson is section 12.1 (EGCMF1201) of our free GCSE Maths Foundation course. In the full course you get every video for this section, an animated interactive lesson, a self-marking practice quiz, a printable worksheet with a mark scheme, and your progress saved. Go to the full GCSE Maths Foundation course.

 

 

 

 

 

 

 

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