Lingley 8 Math

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Pythagoras and his Theorem

Pythagoras of Samosa short history

Pythagoras is arguably one of the most important mathematicians of his time. He was born in 569 BCE on the small Greek island, Samos. During his life, he perfected his method of traveling education, where he taught in Middle-Eastern cities. !Many people of the time could not follow the intricate math theroems, and unfortunately thought that Pythagoras was crazy! Even if some questioned his sanity, Pythagoras attracted like-minded individuals where they continued to learn his teachings in secret as to not

be considered evil. This secret group called themselves The Pythagoreans. Their meetings became so secret,

that they developed their own language, and even had their own seal, etched above the doors where their

meetings were held.

The Pythagoreans thought that all problems could be solved by numbers. This must be how they discovered how to correctly build their iconic Greek columns. Before news of Pytagoras’ Theorem spread, all buildings were formed with crooked bottoms, since there was no tool of measurement to ensure that their bases were straight.

!The Pythagoreans traveled throughout Greece with their special measurement tool: the 12 knot rope. With this they were able to solve many building mistakes.

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Solving the TheoremIf

only these columns were straight!

Understanding the Pythagorean Theorem

Pythagoras saw that the crooked columns casted a triangular shadow on the ground. Using his knowledge of geometry, he saw that the crooked columns casted an acute triangle. He then discovered that the only triangle that will work with his theorem is a right angle triangle.

hypotenuse

leg

right angle

isosceles acute right

x x ✓

Once Pythagoras switched to only using the right angle triangle, he soon found a relationship between the legs on either side of the right angle, and the hypotenuse.

The Secrets behind the Theorem

1. Squares can be formed around each side of the triangle.

2. The sum of the small and medium square areas’ equals the area around the hypotenuse.

3. This relationship is only true for right angle triangles.

4. The theorem is used when solving for an unknown length of a triangle.

5. The theorem will also work for any regular polygon around the sides of the triangle.

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Applying the TheoremUsing the Pythagorean Theorem, find the value of the hypotenuse.

Example 1

6 cm 8 cm

h

h

8 cm

8 cm

6 cm

6 cm

h

1. Draw squares around all sides of the right angle triangle, and label them.

2. Find the areas of each of those squares.

6 cm

6 cm

36 cm2 8 cm

8 cm

64 cm2 + = 100 cm2

3. We now know that the area of the purple square is 100 cm2. However this is only the area! We now need to take the square root to find the side length!

4. Take the square root of the hypotenuse √100 cm2 = 10 cm

Final Answer!

Your Turn! Using the Pythagorean Theorem, find the missing values of the triangles below.

4 cm 4 cm

h

5 cm

10 cm h

6 cm

x

9 cm

Do your work on the next page.

If you find the area is a non-square... approximate!

Lingley 8 Math

Do your work here!