Grade 9 Theorem of Pythagoras

Grade 9 Theorem of Pythagoras: finding the hypotenuse or a shorter side, testing for right-angled triangles, and real-world word problems, with examples.

The Theorem of Pythagoras says that in a right-angled triangle, the square of the hypotenuse equals the sum of the squares of the other two sides: c² = a² + b². The hypotenuse is always the longest side, opposite the right angle.

In Grade 9 you use the theorem to find a missing side, to test whether a triangle has a right angle, and to solve problems such as ladders against walls or the diagonal of a screen. It's used constantly in Grades 10 to 12, in trigonometry, analytical geometry and measurement.

Key concepts

  • c² = a² + b², where c is the hypotenuse
  • Finding the hypotenuse: c = √(a² + b²)
  • Finding a shorter side: a = √(c² − b²)
  • Converse: if c² = a² + b², the triangle is right-angled
  • Pythagorean triples such as 3; 4; 5 and 5; 12; 13

Worked examples

A 5 m ladder leans against a wall with its foot 3 m from the wall. How high up the wall does it reach?

The ladder is the hypotenuse. h² = 5² − 3² = 25 − 9 = 16, so h = 4 m.

Find the diagonal of a 9 cm by 12 cm rectangle.

d² = 9² + 12² = 81 + 144 = 225, so d = 15 cm.

Is a triangle with sides 6 cm, 7 cm and 9 cm right-angled?

Test the longest side: 9² = 81, but 6² + 7² = 36 + 49 = 85. Since 81 ≠ 85, it is not right-angled.

A TV screen is 80 cm wide and 60 cm high. What is its diagonal size?

d² = 80² + 60² = 6 400 + 3 600 = 10 000, so d = 100 cm.

A right-angled triangle has a hypotenuse of 13 cm and one side of 5 cm. Find the third side.

Subtract, because the hypotenuse is known: a² = 13² − 5² = 169 − 25 = 144, so a = 12 cm.

Common mistakes

  • Adding the squares when you need to subtract to find a shorter side.
  • Using a side that isn't the hypotenuse as c.
  • Forgetting to take the square root at the end.
  • Using the theorem on a triangle that isn't right-angled.

Exam tips

  • Identify the hypotenuse first: it is opposite the right angle.
  • Draw and label a sketch for every word problem.
  • Round only at the end, to the accuracy the question asks for.

Frequently asked questions

Does Pythagoras work for every triangle?

No, only right-angled triangles. For other triangles you'll use the sine and cosine rules in Grade 11.

What is a Pythagorean triple?

Three whole numbers that fit c² = a² + b², such as 3; 4; 5, 6; 8; 10 and 5; 12; 13.

How do I know whether to add or subtract?

Add the squares when you're finding the hypotenuse (the longest side). Subtract when you know the hypotenuse and need a shorter side.

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