Calculate the Radius of a Circle with an Inscribed Triangle

Problem Overview

• 🎯 The video demonstrates how to calculate the radius (R) of a circle.
• 💡 A triangle ABC is fully inscribed within the circle.
• 📏 Key given information includes side length BC = 6 units and angle BAC = 60 degrees.

Method 1: Central Angle and Trigonometry

• 🧠 The Central Angle Theorem was applied, stating that the angle at the center (BOC) is twice the angle at the circumference (BAC), resulting in angle BOC = 120 degrees.
• 📐 It was observed that triangle BOC is isosceles because BO and OC are both radii (R).
• ✅ The perpendicular bisector theorem was used by dropping a perpendicular (OP) from the center to chord BC, which bisects BC into two 3-unit segments (PC=BP=3) and bisects angle BOC, creating a 60-degree angle (COP).
• 🔬 The sine trigonometric ratio was then applied in the right triangle CPO: sin(60°) = Opposite/Hypotenuse = 3/R.
• 🔢 Solving for R yielded: R = 3 / sin(60°) = 3 / (√3/2) = 2√3 units.

Method 2: Extended Sine Rule

• 🔑 This method utilized the extended sine rule for a triangle inscribed in a circle, which states: a / sin(alpha) = 2R.
• 📊 The side 'a' was identified as BC (6 units), and the angle 'alpha' as BAC (60 degrees).
• 🧮 Substituting these values into the rule gave: 6 / sin(60°) = 2R.
• ✨ Isolating R resulted in: R = 6 / (2 * sin(60°)) = 3 / sin(60°) = 2√3 units.

Final Radius Calculation

• 👏 Both distinct methods consistently produced a radius of 2√3 units for the circle.
• 📈 The approximate numerical value of the calculated radius is 3.464 units.