UK K12 GCSE/A-Level: Year 8 KS3 Mathematics - Pythagoras' Theorem

Learning Objectives

By the end of this topic, students will be able to:

• Recall and apply Pythagoras' Theorem to solve problems involving right-angled triangles.
• Understand the concept of a right-angled triangle and the relationship between its sides.
• Use the theorem to calculate the length of a missing side in a right-angled triangle.
• Solve problems involving real-world applications of Pythagoras' Theorem.
• Recognize and explain the limitations of Pythagoras' Theorem.

Core Concepts

Pythagoras' Theorem states that in a right-angled triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides. This can be expressed mathematically as:

a² + b² = c²

where a and b are the lengths of the two sides that form the right angle, and c is the length of the hypotenuse.

A right-angled triangle is a triangle with one angle that is 90 degrees. The sides of a right-angled triangle can be classified as:

• Hypotenuse: the side opposite the right angle
• Legs: the two sides that form the right angle

Worked Examples

Example 1: Finding the length of the hypotenuse

A right-angled triangle has one leg that is 5 cm long and the other leg that is 12 cm long. Find the length of the hypotenuse.

Using Pythagoras' Theorem:

a² + b² = c² 5² + 12² = c² 25 + 144 = c² 169 = c² c = ?169 c = 13 cm

Example 2: Finding the length of a leg

A right-angled triangle has a hypotenuse that is 15 cm long and one leg that is 9 cm long. Find the length of the other leg.

Using Pythagoras' Theorem:

a² + b² = c² 9² + b² = 15² 81 + b² = 225 b² = 225 - 81 b² = 144 b = ?144 b = 12 cm

Common Misconceptions

• Many students mistakenly think that Pythagoras' Theorem only applies to triangles with integer side lengths. However, the theorem is applicable to all right-angled triangles, regardless of the lengths of their sides.
• Some students may confuse the concept of a right-angled triangle with other types of triangles, such as isosceles or equilateral triangles.

Exam Tips

• Make sure to read the question carefully and identify the type of triangle involved.
• Check if the triangle is right-angled by looking for the right angle symbol (90°).
• Use Pythagoras' Theorem to find the length of the missing side, and then check your answer by plugging it back into the theorem.
• Be careful when dealing with decimals or fractions, as these can affect the accuracy of your answer.

MCQs with Explanations

MCQ 1: [F]

In a right-angled triangle, the length of the hypotenuse is always greater than the length of the other two sides. Which of the following is a correct statement?

A) The length of the hypotenuse is always equal to the length of the other two sides. B) The length of the hypotenuse is always greater than the length of the other two sides. C) The length of the hypotenuse is always less than the length of the other two sides. D) The length of the hypotenuse is always equal to the sum of the lengths of the other two sides.

Correct answer: B) The length of the hypotenuse is always greater than the length of the other two sides. Why the distractors fail: A) is incorrect because the length of the hypotenuse is not always equal to the length of the other two sides. C) is incorrect because the length of the hypotenuse is always greater than the length of the other two sides. D) is incorrect because the length of the hypotenuse is not always equal to the sum of the lengths of the other two sides.

MCQ 2: [H]

A right-angled triangle has one leg that is 8 cm long and the other leg that is 15 cm long. Find the length of the hypotenuse using Pythagoras' Theorem.

A) 17 cm B) 20 cm C) 21 cm D) 25 cm

Correct answer: B) 20 cm Why the distractors fail: A) is incorrect because 17² + 8² does not equal 20². C) is incorrect because 21² is greater than 20². D) is incorrect because 25² is greater than 20².

MCQ 3: [F]

A right-angled triangle has a hypotenuse that is 10 cm long and one leg that is 6 cm long. Find the length of the other leg using Pythagoras' Theorem.

A) 4 cm B) 6 cm C) 8 cm D) 10 cm

Correct answer: C) 8 cm Why the distractors fail: A) is incorrect because 6² + 4² does not equal 10². B) is incorrect because 6² + 6² does not equal 10². D) is incorrect because 10² is the length of the hypotenuse, not one of the legs.

MCQ 4: [H]

A right-angled triangle has one leg that is 12 cm long and the other leg that is 16 cm long. Find the length of the hypotenuse using Pythagoras' Theorem.

A) 20 cm B) 22 cm C) 24 cm D) 26 cm

Correct answer: B) 22 cm Why the distractors fail: A) is incorrect because 20² is less than 12² + 16². C) is incorrect because 24² is greater than 12² + 16². D) is incorrect because 26² is greater than 12² + 16².

MCQ 5: [F]

A right-angled triangle has a hypotenuse that is 12 cm long and one leg that is 9 cm long. Find the length of the other leg using Pythagoras' Theorem.

A) 5 cm B) 7 cm C) 8 cm D) 10 cm

Correct answer: C) 8 cm Why the distractors fail: A) is incorrect because 9² + 5² does not equal 12². B) is incorrect because 9² + 7² does not equal 12². D) is incorrect because 9² + 10² does not equal 12².

Short-answer Questions

1. A right-angled triangle has one leg that is 8 cm long and the other leg that is 15 cm long. Find the length of the hypotenuse using Pythagoras' Theorem.

(Answer should be 17 cm)

1. A right-angled triangle has a hypotenuse that is 10 cm long and one leg that is 6 cm long. Find the length of the other leg using Pythagoras' Theorem.

(Answer should be 8 cm)

1. A right-angled triangle has one leg that is 12 cm long and the other leg that is 16 cm long. Find the length of the hypotenuse using Pythagoras' Theorem.

(Answer should be 20 cm)

1. A right-angled triangle has a hypotenuse that is 12 cm long and one leg that is 9 cm long. Find the length of the other leg using Pythagoras' Theorem.

(Answer should be 8 cm)

1. A right-angled triangle has one leg that is 15 cm long and the other leg that is 20 cm long. Find the length of the hypotenuse using Pythagoras' Theorem.

(Answer should be 25 cm)