Mastering the 45-45-90 Special Right Triangle

The Organic Chemistry Tutor

This video serves as a comprehensive tutorial on the properties and applications of the 45-45-90 special right triangle, a fundamental concept in geometry and trigonometry. The narrator systematically explains the invariant ratio of the sides (x : x : x√2) and provides clear rules for calculating missing side lengths: multiplying leg lengths by the square root of 2 to find the hypotenuse, and dividing the hypotenuse by the square root of 2 to find the legs. The video covers a progression of difficulty, moving from simple integer side lengths to complex problems requiring the rationalization of denominators. Key mathematical themes explored include the properties of isosceles right triangles, operations with radicals (square roots), and area calculations. The instructor places special emphasis on the algebraic skills necessary to handle radicals, demonstrating step-by-step how to simplify expressions like 10/√2. The video culminates in a challenge problem involving a circle and an inscribed triangle, connecting abstract geometry concepts to the type of multi-step application problems frequently seen on standardized tests like the SAT and ACT. For educators, this video offers a versatile tool for both introducing new concepts and reviewing procedural fluency. It is particularly useful for scaffolding instruction; teachers can pause after each distinct type of example (integer legs, radical legs, integer hypotenuse) to check for understanding. The final problem serves as an excellent extension task, demonstrating how basic triangle properties apply to more complex composite figures involving circles.