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# Pythagorean Theorem a² + b² = c² "In a right triangle, the square of the hypotenuse equals the sum of the squares of the other two sides." Notation (as shown): Legs: a, b | Hypotenuse: c (longest) | Right angle: ∠C = 90° ## 1 What It Is In any right triangle △ABC (∠C = 90°): a² + b² = c². Knowing two sides gives the third. This links length to area. ## 2 Visual Proof (Area View) Large square area = 4×(½ab) + c² = 2ab + c². Also = (a+b)² = a² + 2ab + b². Thus 2ab + c² = a² + 2ab + b² → a² + b² = c². ## 3 Geometric Intuition The square on the hypotenuse equals the sum of squares on the two legs. Think: the two leg-squares exactly tile the hypotenuse-square. ## 4 Numerical Example a=3, b=4: c²=9+16=25 → c=5. (3-4-5 triple) More: (5,12,13), (8,15,17), (7,24,25)... ## 5 Applications Measurement, construction, navigation, computer graphics (Euclidean distance), astronomy. ## 6 Converse (Right-Triangle Test) If a²+b²=c² (c longest), the triangle is right-angled. Example: 6,8,10 → 36+64=100=10² ✓. ## Summary Formula: a²+b²=c² | Use: find sides, test right angles | Key idea: area balance. ## Extensions 3D: d²=a²+b²+c² | Distance formula: PQ = √[(x₂-x₁)²+(y₂-y₁)²]. One sentence: In a right triangle, the hypotenuse squared equals the sum of the legs squared.

Pythagorean Theorem: Right Triangle Side Length Relationship and Visual Area Proof

@Meso Happ (Meso)

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# Pythagorean Theorem a² + b² = c² "In a right triangle, the square of the hypotenuse equals the sum of the squares of the other two sides." Notation (as shown): Legs: a, b | Hypotenuse: c (longest) | Right angle: ∠C = 90° ## 1 What It Is In any right triangle △ABC (∠C = 90°): a² + b² = c². Knowing two sides gives the third. This links length to area. ## 2 Visual Proof (Area View) Large square area = 4×(½ab) + c² = 2ab + c². Also = (a+b)² = a² + 2ab + b². Thus 2ab + c² = a² + 2ab + b² → a² + b² = c². ## 3 Geometric Intuition The square on the hypotenuse equals the sum of squares on the two legs. Think: the two leg-squares exactly tile the hypotenuse-square. ## 4 Numerical Example a=3, b=4: c²=9+16=25 → c=5. (3-4-5 triple) More: (5,12,13), (8,15,17), (7,24,25)... ## 5 Applications Measurement, construction, navigation, computer graphics (Euclidean distance), astronomy. ## 6 Converse (Right-Triangle Test) If a²+b²=c² (c longest), the triangle is right-angled. Example: 6,8,10 → 36+64=100=10² ✓. ## Summary Formula: a²+b²=c² | Use: find sides, test right angles | Key idea: area balance. ## Extensions 3D: d²=a²+b²+c² | Distance formula: PQ = √[(x₂-x₁)²+(y₂-y₁)²]. One sentence: In a right triangle, the hypotenuse squared equals the sum of the legs squared.

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