Pythagorean Theorem


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We learned that in a right triangle the sum of the squares of the legs of the right triangle is equal to the square of the hypotenuse of the right triangle.

The ancient Egyptians used to form right angles by takinng a rope with 12 equally spaced knots. They would form a triangle with sides of length of 3, 4 anf 5 knots. We know that , so does this prove that the triangle they formed was a right triangle?

Activity: Open the GSP sketch and record the values of and if the triangle is acute,obtuse or right in your notebook.

What conjectures can you make?

Pythagorean Thereom converse:

If the square of one side of a triangle is equal to the sum of the squares of the other two sides, then the triangle is a right triangle.

Write the Converse Theorem and complete the proof in your Journal.

Given: ABC with

Prove: ABC is a right triangle.


Outline of proof:


1. Draw a right triangle EFG with legs a and b

2. (why?)

3. (Given)

4.c = n (why?)

5.ABC EFG (SSS postulate)

6. C is a right triangle (why?)

7. ABC is a right triangle (why?)

So when the Egyptians formed a triangle with equally spaced knots as described above, they did form a right triangle!

Pythagorean Triples:

The triangle with sides 3 units,4 units, 5 units forms a right triangle since . Any triple of integers such that is called a Pythagorean triple. It is called a primitive Pythagorean triple if 1 is the only common factor to all three integers.

Tests for Acute, Obtuse or Right Triangles:

In triangle ABC, if c is the longest side of the triangle, then

Relations of sides 

 Type of Triangle







Example: Determine if the triangles with the following sides are acute,right, or obtuse:

 a. 9,40,41

 b. 6,7,8

 Solution:  Solution

 The longest side is 41, so we will comare with .

So our triangle is right.

The longest side is 8, so we will compare with .

So our triangle is acute.


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