Pythagorean Triple Explanation

Topic: Understanding Pythagorean Triples

Grade Level: 5

Duration: 45 minutes

Learning Objectives: By the end of this lesson, students should be able to:

1. Define and understand the Pythagorean theorem.
2. Identify Pythagorean triples and explain their significance.
3. Solve simple problems using Pythagorean triples.

Learning Objectives: By the end of this lesson, students should be able to:

1. Define and understand the Pythagorean theorem.
2. Identify Pythagorean triples and explain their significance.
3. Solve simple problems using Pythagorean triples.

Materials Needed:

• Whiteboard and markers
• Chalkboard and chalk (or presentation equipment)
• Pythagorean Triple examples on handouts or projector
• Rulers or measuring tools
• Exercise sheets

Content

A Pythagorean triple is a set of three whole numbers that satisfy the Pythagorean theorem, which is a fundamental concept in geometry. The theorem states that in a right-angled triangle (a triangle with one 90-degree angle), the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides.

In simple terms, a Pythagorean triple consists of three numbers that can be the lengths of the sides of a right-angled triangle, and they follow this rule:

a² + b² = c²

Where ‘a’ and ‘b’ are the lengths of the two shorter sides of the triangle (the legs), and ‘c’ is the length of the longest side (the hypotenuse). If these numbers satisfy this equation, they form a Pythagorean triple. For example, 3, 4, and 5 is a Pythagorean triple because 3² + 4² = 5² (9 + 16 = 25).

Common Pythagorean triples include 3, 4, 5 and 5, 12, 13. These triples are used in mathematics and engineering to solve various problems related to right triangles.

1. Example 1:
   Given a right-angled triangle with one leg of length 6 units and the other leg of length 8 units, you can check if it’s a Pythagorean triple using the Pythagorean theorem:

   a = 6, b = 8, c² = a² + b² c² = 6² + 8² c² = 36 + 64 c² = 100 c = √100 c = 10

   So, the Pythagorean triple is 6, 8, 10 because 6² + 8² = 10².
2. Example 2:
   Given a right-angled triangle with one leg of length 5 units and the hypotenuse of length 13 units:

   a = 5, c = 13, b² = c² – a² b² = 13² – 5² b² = 169 – 25 b² = 144 b = √144 b = 12

   So, the Pythagorean triple is 5, 12, 13 because 5² + 12² = 13².
3. Example 3:
   Given a right-angled triangle with one leg of length 9 units and the other leg of length 12 units:

   a = 9, b = 12, c² = a² + b² c² = 9² + 12² c² = 81 + 144 c² = 225 c = √225 c = 15

   So, the Pythagorean triple is 9, 12, 15 because 9² + 12² = 15².
4. Example 4:
   Given a right-angled triangle with one leg of length 7 units and the hypotenuse of length 25 units:

   a = 7, c = 25, b² = c² – a² b² = 25² – 7² b² = 625 – 49 b² = 576 b = √576 b = 24

   So, the Pythagorean triple is 7, 24, 25 because 7² + 24² = 25².
5. Example 5:
   Given a right-angled triangle with one leg of length 20 units and the other leg of length 21 units:

   a = 20, b = 21, c² = a² + b² c² = 20² + 21² c² = 400 + 441 c² = 841 c = √841 c = 29

   So, the Pythagorean triple is 20, 21, 29 because 20² + 21² = 29².

1. A Pythagorean triple with sides 20, 21, and 29 is an example of a(n) ________. a) Equilateral triangle b) Isosceles triangle c) Right-angled triangle d) Scalene triangle
2. The Pythagorean triple 5, 12, 13 can be used to describe the sides of a ________ triangle. a) Equilateral b) Isosceles c) Right-angled d) Acute
3. The Pythagorean theorem is used to find the lengths of sides in ________ triangles. a) Equilateral b) Isosceles c) Right-angled d) Obtuse
4. The Pythagorean triple 8, 15, 17 satisfies the equation ________. a) 8² + 15² = 17² b) 17² + 15² = 8² c) 8 + 15 = 17 d) 8² – 15² = 17²
5. A Pythagorean triple can only be formed with ________ numbers. a) Prime b) Irrational c) Whole d) Negative
6. The Pythagorean triple 6, 8, 10 is an example of a ________ Pythagorean triple. a) Primitive b) Quadratic c) Trigonometric d) Complex
7. A Pythagorean triple is a set of three numbers that can represent the sides of a ________ triangle. a) Right-angled b) Isosceles c) Equilateral d) Scalene
8. Which of the following is NOT a Pythagorean triple? a) 9, 12, 15 b) 5, 12, 13 c) 6, 8, 10 d) 2, 3, 5
9. The Pythagorean triple 15, 17, 22 is an example of a(n) ________ Pythagorean triple. a) Primitive b) Irregular c) Impossible d) Invalid
10. A Pythagorean triple is a special case of the Pythagorean theorem, where the sides are all ________ numbers. a) Rational b) Irrational c) Decimal d) Fractional
11. In a Pythagorean triple, the sum of the squares of the two shorter sides is equal to the square of the ________. a) Hypotenuse b) Perimeter c) Area d) Altitude
12. A Pythagorean triple is a set of three whole numbers that satisfy the Pythagorean ________. a) Square b) Addition c) Division d) Multiplication
13. The Pythagorean triple 7, 24, 25 follows the Pythagorean theorem because 7² + 24² = ________. a) 25² b) 49 c) 576 d) 576²
14. What is the Pythagorean triple for which a = 9, b = 12, and c = ________? a) 6 b) 15 c) 21 d) 10
15. In the Pythagorean triple 3, 4, 5, what is the length of the hypotenuse? a) 3 b) 4 c) 5 d) 7

Introduction (10 minutes):

1. Begin the lesson by asking students if they have ever heard of the Pythagorean theorem.
2. Write the Pythagorean theorem on the board: a² + b² = c² and briefly explain that it relates to right-angled triangles.
3. Define the terms: a (one leg), b (the other leg), and c (the hypotenuse).

Main Lesson (25 minutes):

1. Present Pythagorean triples as sets of three whole numbers that satisfy the Pythagorean theorem.
2. Explain that common Pythagorean triples include 3, 4, 5 and 5, 12, 13.
3. Show examples of right-angled triangles and demonstrate how to determine if they are Pythagorean triples.
4. Work through several examples on the board and encourage student participation.
5. Discuss the practical applications of Pythagorean triples in real life, such as measuring distances, constructing right angles, and more.

Group Activity (10 minutes):

1. Divide the class into small groups.
2. Provide each group with a set of three numbers and ask them to determine if they form a Pythagorean triple.
3. Circulate the classroom to assist and check their work.

Conclusion (5 minutes):

1. Have each group share their findings with the class.
2. Recap the key points of the lesson, including the definition of Pythagorean triples and their application.
3. Assign homework or additional practice problems related to Pythagorean triples.

Assessment: Assess students based on their participation in group activities, their ability to identify Pythagorean triples, and their understanding of the concept.

Homework: Assign exercises for students to practice identifying Pythagorean triples and using the Pythagorean theorem to solve simple problems.

Extension: For advanced students, introduce the concept of primitive Pythagorean triples and explain how to generate new triples using primitive ones.

Note: Always adapt the lesson plan to the specific needs and pace of your class, providing additional explanations or examples as necessary.