The Pythagorean theorem
The hypotenuse and the legs, Pythagorean triples and answers with roots, the distance between points and the diagonal of a box, a ladder against a wall — from simple to two-step problems.
How to solve it
In a right triangle the square of the hypotenuse is the sum of the squares of the legs. The hypotenuse is the side opposite the right angle, and it is the longest. The distance between two points in the plane is a hypotenuse too: the legs are the differences of the coordinates.
Step by step
- Find the right triangle in the problem and which side is the hypotenuse: the ladder, the diagonal, the segment between the points.
- The hypotenuse: $c = \sqrt{a^2 + b^2}$. A leg: $a = \sqrt{c^2 - b^2}$.
- If the root is not exact, simplify it: $\sqrt{68} = 2\sqrt{17}$.
- The diagonal of a box takes two steps: first the diagonal of the bottom, then the triangle it makes with the height. In total $d = \sqrt{a^2 + b^2 + c^2}$.
- Remember the common triples: $3, 4, 5$; $5, 12, 13$; $8, 15, 17$; $7, 24, 25$; $20, 21, 29$; $11, 60, 61$.
Common mistakes
- Adding the squares when looking for a leg: a leg is the root of the difference, $\sqrt{c^2 - b^2}$.
- Taking the root of each term: $\sqrt{8^2 + 2^2} \ne 8 + 2$.
- Not simplifying the root: $\sqrt{68}$ is $2\sqrt{17}$.
- In “the ladder slid away” subtracting the moves of the foot instead of the heights: compute the height before and after, then subtract.