The Pythagorean theorem

前置知识

11 项必需的前置知识（一直追溯到底）

你将学会

• Explain at least one proof of the Pythagorean theorem, such as a rearrangement proof.
• Use the theorem and its converse to find missing lengths and to test whether a triangle is right-angled.
• Compute the distance between two points in the coordinate plane.

定义

• A right triangle has one angle of 90°; the side opposite it is the hypotenuse.
• For x ≥ 0, the square root √x is the non-negative number whose square is x.

定理

• Pythagorean theorem: in a right triangle with legs a, b and hypotenuse c, a² + b² = c².
• Converse: if the sides of a triangle satisfy a² + b² = c², the triangle has a right angle opposite the side c.
• The distance between (x₁, y₁) and (x₂, y₂) is √((x₂ − x₁)² + (y₂ − y₁)²).
• There are infinitely many Pythagorean triples, such as (3, 4, 5) and (5, 12, 13).

常见误解

• Applying a² + b² = c² to triangles that are not right-angled.
• Taking c to be any side rather than the hypotenuse.
• Thinking √(a² + b²) = a + b.

通向

可选发现

• Real numbers and irrationality A square of side 1 has a diagonal of length √2, which cannot be written as a fraction of whole numbers.
• Number theory The primitive Pythagorean triples are exactly (m² − n², 2mn, m² + n²), up to swapping the first two, with m > n > 0 coprime and not both odd. For n ≥ 3, xⁿ + yⁿ = zⁿ has no solutions in positive integers (Fermat's Last Theorem, proved by Wiles in 1995).

学习路径

• Conventional K–12 progression

延伸阅读

• Euclid, Elements — Book I, Proposition 47.