The Pythagorean Theorem — Rearranging Pieces to a²+b²=c²
Fill one big square two ways with four identical right triangles: once leaving a² and b², once leaving c². Equal leftovers ⟹ a²+b²=c².
Do not write an equation yet. Shake the figure first.
Play the reversal slowly
Now move the same pieces step by step and prove that the pattern you touched was not an accident.
A square of side a+b. Four equal right triangles leave gaps forming two squares, a² and b².
Return to the exact question
Prove a² + b² = c² for a right triangle by rearranging the same four triangles inside one square.
Why this approach works
Don’t compute areas — compare two arrangements of the same pieces. In a square of side a+b, place four right triangles so the gaps form two squares a² and b². Translate those four triangles into the four corners, and the gap becomes one square c² on the hypotenuse. Same big square, same four triangles, so the leftovers are equal: a² + b² = c². Algebraically, (a+b)² − 4·½ab = c².
Proof
Comparing two ways to fill an (a+b)-square with the same four right triangles: one leaves a²+b², the other leaves c². The big square and triangles are identical, so the leftovers are equal — a²+b²=c².
Try it yourself
Subtract the four triangles (each ½ab) from (a+b)²: the 2ab cancels, leaving a²+b² — which equals c².
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