Grades 6–8Geometry & TopologyPlanned
The Pythagorean theorem
The relation a² + b² = c² for right triangles, proofs by rearrangement, the converse, square roots, and distances in the coordinate plane.
math.geom.pythagorean-theorem
The lesson for this topic has not been written yet. The outline below is part of the curriculum map and has not been reviewed.
Builds on
Required prerequisites, all the way down: 11
You will learn to
- 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.
Definition
- 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.
Theorem
- 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).
Common misconceptions
- 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.
Leads to
Optional discoveries
- 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).
Learning pathways
Further reading
- Euclid, Elements — Book I, Proposition 47.