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Key Terms Inscribed Polygon Inscribed right Triangle-Diameter Theorem Inscribed Right Triangle-Diameter Converse Theorem Circumscribed Polygon Inscribed Quadrilateral-Opposite Angles Theorem http://www.youtube.com/watch?v=bbxO4HdCW5A An inscribed polygon is a polygon drawn inside a circle such that each vertex of the polygon touches the circle. A circumscribed polygon is a polygon drawn outside a circle such that each side of the polygon is tangent to a circle. pg 724 versus pg 728 You just completed the outline for a proof! Inscribed Right Triangle-Diameter Theorem- “If a triangle is inscribed In a circle such that one side of the triangle is a diameter of the circle, then The triangle is a right triangle.” pg 725-726 The Inscribed Right Triangle-Diameter Converse Theorem- “If a right triangle is inscribed in a circle, then the hypotenuse is a diameter of the circle. Notice now you are given a right triangle, hence a right angle. By in Inscribed Angle Theorem you then know the arc opposite is twice the measure, hence 180˚. By definition, 180˚is a semicircle hence, the hypotenuse is a diameter. pg 727 The Inscribed Quadrilateral-Opposite Angles theorem- “If a quadrilateral is inscribed in a circle, then the opposite angles are supplementary.” A circle measures 360˚, so arc QUA arc QDA 360 We also know the measures of the associated angles are ½ So QUA QDA 180 pg 730 The point at which you place your compass to drawn the circle is called the in-center and is equidistant to the sides of the polygon. Notice the distance to the side is the radius of the circle and also tangent. To extend this idea to other shapes we do the exact same thing, find the angle bisectors, find the in-center, use the in-center as the center of the circle to be inscribed. pg 731