Solve This Advanced #Geometry Puzzle | Area of the Blue Region Explained 🔵📐

🔵 Welcome back to Prime Logic! In today's deep dive, we're tackling an advanced and exciting geometry puzzle: 🧠 A right triangle with a semicircle, a small inscribed circle, and lots of clever trigonometry tricks! We'll use Pythagoras' theorem, tangent double angle formulas, and some algebra to find the area of the mysterious blue region. This video walks you through the solution step-by-step in around 18 minutes. If you love intuitive explanations, geometry challenges, and sharpening your math skills, you're in the right place! 📝 Topics Covered: Geometry with semicircles and tangent circles Trigonometric identities (tan(2x)) Solving quadratic equations in geometry Pythagorean theorem applications Deep intuition behind the formulas — 💬 Drop your thoughts or your solutions in the comments — I learn a lot from you too! Let's build an awesome math community together. 🚀 Subscribe for more geometry puzzles, math challenges, and problem-solving tricks every week! ➡️ Don't forget to Like 👍, Share 📤, and Subscribe 🔔 to keep Prime Logic growing! #MathPuzzle #GeometryChallenge #PrimeLogic #ProblemSolving #LearnMath #MathTutorial #GeometrySolutions ⏱️ Timestamps 00:00 - Introduction to the Blue Area Puzzle 00:40 - Problem Setup and Strategy 02:00 - Understanding the Triangle and Semicircle Relationship 04:00 - First Step: Finding Segment 'b' with Pythagoras 06:10 - Solving for 'b' (Calculation Walkthrough) 08:00 - Working on Segment 'a' and Congruent Triangles 10:00 - Applying the Double Angle Formula for Tangent 12:30 - Setting Up and Solving the Equation for 'a' 14:30 - Finding the Height of the Triangle 16:00 - Calculating the Final Blue Area 17:30 - Final Thoughts and Community Interaction 18:00 - Outro and Thank You! 📢 Call to Action: 👍 Like, 💬 Comment, & 🔔 Subscribe: If you enjoyed this deep dive into advanced geometry, hit the like button, share your alternative solutions or questions in the comments, and subscribe for more mind-bending math challenges. 📤 Share: Spread the knowledge with fellow math lovers and challenge them to solve this puzzle! 🤝 Engage: Follow us on social media for extra content, behind-the-scenes insights, and updates on our latest math puzzles. 🔗 Useful Links: Subscribe to Prime Logic: https://www.youtube.com/@UCLEXLDx0V0hdhRblDUXRWCg 🔗 More Olympiad Challenges Playlist: https://www.youtube.com/watch?v=ue2kKWt6LOs&list=PLBLtQgz8oB1dRygp711jja0rozJvW_jOH 🔗 Connect on Social Media: 🐦 Twitter: https://x.com/PrimeLogics 📸 Instagram: https://www.instagram.com/primelogic/