Four triangles and one vacancy
Four identical guests occupy a square hotel. The empty room solves the mystery.
The formula in this story
Story 7 of 8
What you will take away
Explain why removing four triangles leaves area a² + b².
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What do the words mean?
A right triangle has one 90° angle. Its hypotenuse c is opposite that angle; a and b are the two legs. An acute angle is less than 90°.
Want a hand with the prerequisites?The missing two cake strips · The triangle that lost a corner
A suspicious plan
Pip builds a square hotel of side a + b and rents four corners to identical right triangles.
Four congruent right triangles leave a square: its sides are c and its angles are right angles.
The clue
Their hypotenuses enclose a room. Mika checks its corners: complementary acute angles leave a right angle each time. The room really is a square of side c.
The large square has area (a + b)². Four triangles occupy 4 × ab/2 = 2ab.
The twist
Count the hotel area, then subtract the four triangle rooms. The vacancy has area a² + b². Pip: “Can empty rooms pay rent?”
Subtract only the four triangles. The central area is c² = (a + b)² − 2ab = a² + b².
Change the evidence
Try examples to build intuition. Checking numbers is not the proof for every value.
Step 1 of 3
Arrange four copies inside a square of side a + b. The remaining region has four sides of length c. Its angles are 90° because the two acute angles of a right triangle sum to 90°.
The diagram illustrates the displayed argument. Numerical examples do not replace its hypotheses or prove every case.
The proof, with all its steps
Claim and conditions
For a Euclidean right triangle with positive legs a, b and hypotenuse c, a² + b² = c².
- Arrange four copies inside a square of side a + b. The remaining region has four sides of length c. Its angles are 90° because the two acute angles of a right triangle sum to 90°.
- The whole area is (a + b)². Each triangle has area ab/2, so the four occupy 2ab.
- Thus c² = (a + b)² − 2ab = a² + b², using the expansion we proved earlier.
Pip has one more idea…
May we conclude a + b = c by taking the square root term by term?
Read the explanation for Yes, square roots distribute over addition.
A square root does not distribute over sums. √(9 + 16) = 5, while √9 + √16 = 7.
Read the explanation for No: for legs 3 and 4, the hypotenuse is 5, not 7.
Exactly. The relation concerns squares. It also supplies a useful contrast with Fermat: exponent 2 does have positive integer solutions.
What makes more sense now?
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These are educational arguments and numeric experiments, not live Lean verification. The linked knowledge pages distinguish formal source excerpts from explanations.