How to Find Missing Triangle Sides: From Pythagoras to Trigonometry

The Organic Chemistry Tutor

This comprehensive mathematics tutorial provides a deep dive into calculating missing side lengths of triangles using various methods suitable for different scenarios. The video progressively builds complexity, starting with fundamental concepts like the Pythagorean Theorem for right triangles, moving into trigonometric ratios (Sine, Cosine, Tangent) for finding sides when angles are known, and advancing to the Law of Sines and Law of Cosines for non-right triangles. This structured approach helps students understand not just how to solve a problem, but how to select the correct tool for the specific geometry they face. The content covers key geometry and algebra themes including right triangle properties, SOH CAH TOA, radical simplification, rationalizing denominators, and solving systems of equations derived from geometric properties. A significant portion of the video is dedicated to analyzing composite triangles—shapes formed by combining multiple triangles—and using the Geometric Mean Theorem (Altitude Rule). The narrator demonstrates two distinct methods for solving altitude problems: applying the direct geometric mean formula and proving the result through a system of equations based on the Pythagorean Theorem, bridging the gap between geometric theory and algebraic proof. For educators, this video is an invaluable resource for teaching high school Geometry and Trigonometry. It effectively models mathematical thinking by showing multiple problem-solving strategies and tackling "messy" answers involving radicals and decimals. Teachers can use specific segments to target individual standards (e.g., just Law of Cosines) or use the later sections to challenge advanced students with multi-step composite problems. The clear, step-by-step narration makes it ideal for flipping the classroom or providing support for students struggling with specific triangle theorems.