TwiddleLab / Math / Shapes and Proof

Pythagoras by Rearranging

Lesson 4 of 10 · 7 min · Free, no account needed

What you will learn

• In a right triangle, a² + b² = c², where c is the long side opposite the right angle.
• Why: the same four triangles in the same square leave the same empty area, c² or a² + b².
• Only for 90°: a wider angle makes c² bigger, a narrower one smaller.

Start the lesson

How the lesson goes

1. Your TV is 50 inches. That's the diagonal. It is 30 inches tall. How wide is it?

   Bet first. Drag the lever.

2. Four copies of a right triangle sit in a 7 by 7 square.

   The tilted hole in the middle is the square on the long side, c². Drag the three loose triangles into the dashed spots.

3. Does it work for every triangle?

   The sides 3 and 4 stay, the corner between them turns. Find the angle where a² + b² = c².

   Opening the corner pushes the ends of the legs apart, so c grows. There is exactly one angle where c² hits 25: the right angle.

4. This is the Pythagorean theorem.

   In a right triangle with legs a, b and long side c (the hypotenuse): a² + b² = c².

   Your TV: 30² + w² = 50². So w² = 2500 − 900 = 1600, and w = 40 inches.

   To find c, take a square root: c = √(a² + b²). To find a leg: a = √(c² − b²).

5. Find a right triangle of your own with whole sides.

   Drag the ends of the legs. The long side has to come out whole, and 3, 4, 5 doesn't count.

   Whole-number right triangles are rare: a² + b² must be a perfect square. Scaling 3, 4, 5 always works, and 5, 12, 13 is a new family.

6. Game: ladder rescue.

   Three cats, three windows. Set how far the ladder's foot is from the wall, then Raise. Fewest misses wins.

7. Games check distances millions of times a second.

   To save time they skip the square root and compare squares. What does this print?

   The points differ by 3 across and 4 up: a right triangle with legs 3 and 4. dx² + dy² = c² = 25.

8. A right triangle has legs 9 and 12. How long is the hypotenuse?

   c² = 9² + 12² = 81 + 144 = 225, so c = 15.

9. Your triple, checked cell by cell.

Previous lesson: Similar Triangles and Shadows · Next lesson: Proof as a Chain · Part of the course: Shapes and Proof

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