The Triangle of Mystery

 

Sammy was a curious boy who loved to explore and discover new things. He had a passion for math, especially geometry. He liked to draw and measure shapes, angles, and distances. He also liked to solve puzzles and riddles, and find patterns and connections.

One day, he found a mysterious map in his attic. It was old and torn, and had a strange symbol on it. It looked like a triangle with a square on each side. It also had some numbers and letters on it.

"Wow, look at this!" Sammy exclaimed. "It's a map of a treasure!"

He examined the map carefully. He saw that it had three locations marked on it: A, B, and C. Each location had a number next to it: 4, 3, and 5. There was also an arrow pointing from A to B, from B to C, and from C to A.

He read the instructions on the map:
"Welcome to the triangle of mystery.

A treasure to find! Follow the map, and leave nothing behind. Seek the old oak, with the triangle mark, And there you'll find the treasure, in the dark.

 1. This is your starting point, and we   call it point A.

2. From A, journey four paces south.

3. From your new position, turn east and travel three paces.

4. To complete your trek, return to your starting point, traveling five paces north-west.
The symbol on the map is the key to the puzzle.
Solve this riddle:

What is the relationship between the three sides of a right triangle?

Good luck, adventurer!"

Sammy was intrigued and excited by the map. He decided to follow it and find the treasure.

He took his compass, his ruler, his pencil, and his notebook with him.
He went outside and looked for a suitable place to start his adventure.

He found a large field with some trees and rocks.He chose a tree as his starting point. He used his compass to find the direction of location A.

He used his ruler to measure 4 units from his starting point. He marked location A with a rock. He repeated the same steps for locations B and C.

He marked them with rocks as well. He connected the rocks with his pencil.

He drew a triangle on the ground. He looked at the triangle and noticed something interesting.

It was a right triangle. A right triangle is a triangle that has one angle that is 90 degrees. A 90 degree angle is also called a right angle.
A right angle is formed when two lines meet at a point and make a square corner.

Sammy checked the angle at location B with his ruler.
It was indeed a right angle.
He wondered what that meant.

He remembered the puzzle on the map:
What is the relationship between the three sides of a right triangle?

He looked at the three sides of the triangle: AB, BC, and CA.
He knew their lengths: 4, 3, and 5 units.
But he did not know their relationship.

He thought hard and tried to find a pattern or a connection between them.

He tried adding them together: 4 + 3 + 5 = 12
But that did not seem to help him.
He tried multiplying them together: 4 x 3 x 5 = 60
But that did not seem to help him either.
He tried dividing them by each other: 4 / 3 = 1.33, 3 / 5 = 0.6, 5 / 4 = 1.25

But that did not seem to help him at all.
He was stuck and frustrated.
He looked at the map again.
He saw the symbol on it: a triangle with a square on each side.
He wondered what it meant.

He decided to draw it on his notebook.

He drew a triangle with three sides: A, B, and C.

He drew a square on each side: AA', BB', CC'.

He labeled the vertices of each square: A', B', C'.

He looked at his drawing and noticed something amazing.

The triangle fit perfectly inside the squares!
The squares formed the right triangle!
Sammy was astonished and delighted by his discovery. He wondered if this had something to do with the puzzle.

He decided to measure the areas of each square.
The area of a square is equal to its side length squared.
For example, if a square has a side length of 2 units, its area is 2 x 2 = 4 square units.

Sammy measured the side lengths of each square on his drawing.

He found that:

The side length of square AA' was equal to the side length of AB, which was 4 units.

The side length of square BB' was equal to the side length of BC, which was 3 units.

The side length of square CC' was equal to the side length of CA, which was 5 units.

He calculated the areas of each square on his drawing.

He found that:

The area of square AA' was 4 x 4 = 16 square units.
The area of square BB' was 3 x 3 = 9 square units.
The area of square CC' was 5 x 5 = 25 square units.

He wrote down his results on his notebook:

AA' = 16

BB' = 9

CC' = 25

He looked at his results and noticed something incredible.

The area of square CC' was equal to the sum of the areas of squares AA' and BB'!

25 = 16 + 9

He wondered if this was a coincidence or a rule.
He decided to test it with different numbers.

He chose different values for the side lengths of AB, BC, and CA.

He made sure that they formed a right triangle.

He drew the squares on each side and calculated their areas.

He added the areas of the squares on the shorter sides and compared them to the area of the square on the longer side.

He found that:

Every time, the area of the square on the longer side was equal to the sum of the areas of the squares on the shorter sides!

He realized that this was not a coincidence. It was a rule.

It was a rule that applied to any right triangle.

It was a rule that described the relationship between the three sides of a right triangle.

It was a rule that solved the puzzle on the map.
It was a rule that he had discovered by himself.
It was a rule that had a name.
It was Pythagoras' theorem.