Investigating Pythagorean Theorem

In the Geogebra activity above, a very special right angled triangle is shown. It has side lengths 3, 4 and 5. A set of three whole number side lengths that satisfies Pythagoreans Theorem like this are called Pythagorean Triples. Use the activity to move the two orange points to create different triangles. Find as many Pythagorean Triples as you can and list them in the text box below.

Review your findings from the previous question. Did you notice any interesting relationships between your sets of Pythagorean Triples? If so, explain your findings below.

What is a vocabulary word that describes the relationship between the four green triangles? Explain how you know your answer is true. (Hint: What theorem can you use?)

What types of triangles are the four green triangles?

Why is the pink space a square?

Write an expression for the area of the pink square in terms of c.

Write an expression for the area of the upper pink square in terms of a.

Write an expression for the area of the lower pink square in terms of b.

Write an expression for the total area of the pink space in terms of a and b.

Observe the animation a few more times by moving the slider from the left to the right. Would you say that pink area becomes larger, smaller, or remains equal as you move the slider from left to right?

Express your answer from the previous question as an equation. (Hint: What were the area functions you found earlier?)

The Pythagorean Theorem states that the side lengths of a right triangle are related by the equation a² + b² = c², where a and b are legs and c is the hypotenuse. Did this inductive reasoning activity help convince you about the truth of the Pythagorean Theorem? Explain.

Enter the value you found for the missing side below. (Hint: Look at the missing side. Is it a LEG or the HYPOTENUSE?)

Enter the value you found for the missing side below. (Hint: Look at the missing side. Is it a LEG or the HYPOTENUSE?)