1. The Pythagorean Relationship
“Chuck Norris can use the pythagorean theorem on ANY type of triangle.”

2. The Right Triangle
To discover what the pythagorean relationship is, we have to take a look at the properties of a right triangle
It’s important to know two terms that relate to the right triangle:
Legs – The two shorter sides of a right
triangle. The two legs meet at the 90°
angle.
Hypotenuse – The longest side of a
right triangle. It is the opposite to the
right angle.

3. What is the Pythagorean Relationship?
Follow the instructions on the handout:
Draw and cut the selected squares from grid paper (15 in total). Write the area on each square piece.
Make a triangle with each set of 3 squares. This will form a triangle. Stick each arrangement in your notes.
Copy and complete the table based on your observations of the five triangles.
Use the reflection questions to find relationships with your results.

4. What is the Pythagorean Relationship?
Reflection Questions:
Consider the first three triangles. Compare the areas of the squares on the sides of each triangle. Write a sentence to describe the relationship among them.
The areas of the two shorter sides add up to the area of the larger side.
Examine the last two triangles. Does the relationship for the first three triangles hold for these two? How are these triangles different from the first three?
The relationship does not exist for these two triangles. The relationship only exists with right triangles. There are no right angles with the last two.

5. The Pythagorean Relationship
The relationship you have explored is called the Pythagorean Theorem. It relates the areas of the squares on the three sides of a right triangle:
Pythagorean Theorem - The area of the
square on the hypotenuse is equal to the
sum of the areas of the squares on the legs.
The relationship can be expressed by the
equation: a² + b² = c²
“c” represents the length of the
hypotenuse.
“a” and “b” represents the lengths of the
legs.
We can use this relationship to find the
length of any side of a right triangle, when
we know the lengths of the other two sides.
The Pythagorean Mystery!

6. Example
Find the length of the unmarked side in each right triangle. Give the length to the nearest millimetre.
The area of the square on the hypotenuse is 13². The areas of the squares on the legs are 12² and b².
Using algebra, I substitute the values into the pythagorean equation, and solve for the missing value (b).
c² = a² + b²
13² = 12² + b²
169 = b²
169 – 144 = 144 – b²
25 = b²
√25 = √b²
5 = b b
13cm
12cm

7. Practice Questions
Find the length of the unknown side in each triangle. Give the lengths to the nearest millimetre. c
6cm
26cm
10cm
8cm
Andrew uses a ruler and compass to construct a triangle with side lengths 3cm, 5cm, and 7cm. Before Andrew constructs the triangle, how can he tell if the triangle will be a right triangle? Explain.
The hypotenuse of a right triangle is 10 units. What are possible lengths of the legs of the triangle? Can you find more than one? Sketch a triangle for each answer.