The Pythagorean theorem
Identify the right angle and hypotenuse, apply the squared-length relationship, and check lengths.
Meaning
In a right triangle, the sum of the squares of the legs equals the square of the hypotenuse. Conversely, if this relationship holds with the longest side as hypotenuse, the triangle is right-angled. Place four congruent right triangles with legs a and b inside a square of side a+b, leaving a central square of side c. Subtract their areas from the total: (a+b)²−4×ab/2=c². Simplifying gives a²+b²=c².
Example
Practice for this concept
Scope and references
Topics
Right-triangle side relationships, hypotenuses, the converse, distances
Learning goal
Identify the right angle and hypotenuse, apply the squared-length relationship, and check lengths.
Helpful background
Connect right angles, hypotenuses, squares, and positive lengths.
Lengths in right triangles
Does not cover the theorem's proof or converse, or all properties of general figures.