$Pythagorean Theorem infographic - Right Triangles, Distance, and Pythagorean Triples$

Click image to open full size

Math

Pythagorean Theorem

Right Triangles, Distance, and Pythagorean Triples

Download PNG Save to Pinterest

Related Tools

Pythagorean Theorem Explorer

Related Labs

Related Worksheets

Related Cheat Sheets

The Pythagorean theorem states that in any right triangle, the square of the hypotenuse (the side opposite the right angle) equals the sum of the squares of the other two sides: $a^2 + b^2 = c^2$ . This simple relationship has been independently discovered and proven hundreds of times across many cultures and is arguably the most famous theorem in mathematics.

Applications extend far beyond triangles. The distance formula in the coordinate plane is a direct application of the theorem: d = √((x₂-x₁)² + (y₂-y₁)²). In three dimensions it extends to d = √(x² + y² + z²). Pythagorean triples - integer solutions like (3, 4, 5) and (5, 12, 13) - appear in construction, navigation, and computer graphics.

Key Facts

Pythagorean theorem: $a^2 + b^2 = c^2$ where $c$ is the hypotenuse.
Common triples: (3,4,5), (5,12,13), (8,15,17), (7,24,25).
Multiples of triples also work: (6,8,10), (9,12,15).
Distance formula: d = √((x₂-x₁)² + (y₂-y₁)²)
Converse: if $a^2 + b^2 = c^2$ , the triangle is a right triangle.
If a² + b² > c², the triangle is acute; if a² + b² < c², it is obtuse.

Vocabulary

Hypotenuse: The longest side of a right triangle, opposite the right angle; labeled c in the theorem.
Legs: The two shorter sides of a right triangle, labeled a and b in the theorem.
Pythagorean triple: A set of three positive integers that satisfy $a^2 + b^2 = c^2$ , such as $(3, 4, 5)$ .
Converse: The reverse statement: if $a^2 + b^2 = c^2$ , then the triangle is a right triangle.
Distance formula: Formula derived from the Pythagorean theorem to find the distance between two points in a coordinate plane.

Common Mistakes to Avoid

Adding the sides instead of the squares: $c \neq a + b$ . You must square first, add, then take the square root.
Identifying the hypotenuse incorrectly. The hypotenuse is always opposite the right angle and is always the longest side.
Using the theorem when the triangle is not a right triangle. The theorem only applies to right triangles.
Forgetting to take the square root at the end: a² + b² gives c², not c. Always take the square root to find the actual length.

Practice Questions

1 A ladder 10 m long leans against a wall. If the base is 6 m from the wall, how high up does the ladder reach?
2 Find the distance between the points (1, 2) and (4, 6) using the distance formula.
3 Is a triangle with sides 9, 40, and 41 a right triangle? Show your work.

Sign in to save