Unlocking the Secrets of Inscribed Polygons

• Inscribed Polygon: A polygon is said to be inscribed in a circle if all its vertices lie on the circle. The circle is then called the circumscribed circle of the polygon.

• Any side of the inscribed polygon is also a chord of the circle.
• The angle subtended by a chord at the center is double the angle subtended by it at any point on the remaining part of the circle. This becomes particularly interesting for inscribed polygons.

• The sum of the opposite angles of any quadrilateral inscribed in a circle is \(180^\circ\).
• In an inscribed triangle, the length of its sides can determine the circle’s radius using certain relationships, such as the circumradius formula.

Examples

Practice Questions:

1. A regular hexagon is inscribed in a circle of radius \(4 \text{ cm}\). What is the length of one side of the hexagon?
2. An equilateral triangle is inscribed in a circle. If one side of the triangle is \(9 \text{ cm}\), can you determine the circle’s radius?

1. Each side of the hexagon is equal to the chord of the circle subtended by \(60^\circ\). Using the formula for chord length: \( c \approx 2 \times 4 \times \sin(30^\circ) = 4 \text{ cm}\).
2. For an equilateral triangle inscribed in a circle with a side length of \(9 \text{ cm} \), by using the Pythagorean theorem on the right triangle formed with the radius and half the triangle’s side, the circle’s radius is determined to be \(3\sqrt{3} \text{ cm} \).

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