Inscribed and Circumscribed Polygons

Inscribed n If all of the vertices of a polygon lie on the circle, then the polygon is inscribed

Circumscribed n We can also describe this inscribed quadrilateral as a circle circumscribed about the quadrilateral

With your partner… n Sketch another large circle with your compass. n Note where the center of the circle is located when you’re sketching the circle, and draw the diameter of the circle. n Sketch a triangle such that the vertices of the triangle lie on the circle, making sure that the diameter is one of the sides of the triangle. n Measure each angle in the triangle and make a note of them.

? n When you sketched the triangle with the diameter being one side of the triangle, what did you find the vertex to be?

There is a theorem that describes this relationship…

O D C B Wherever you place the vertex with the right angle, the hypotenuse must be the diameter. Why do you think this is???

There is a theorem that describes this relationship…

Let’s try and use this theorem! n Solve for x n 15x = 90 0 n x = 6

With your partner… n Sketch a large circle with your compass n Sketch a quadrilateral such that every vertex lies on the circle. n Label the vertices with any letters. n With your protractor, measure the opposite angles, and list them.

When measuring the opposite angles, what did you find them to be?

There is a theorem that states this special property Theorem A quadrilateral can be inscribed in a circle if and only if its opposite angles are supplementary.

Therefore,

Let’s try using Theorem n Find the value of each variable.