10.7: Inscribed and Circumscribed Polygons

1. 10.7: Inscribed and Circumscribed Polygons

2. What is an inscribed polygon? • Def: An inscribed polygon is a polygon with all of its vertices on the circle. M MATH is an inscribed polygon because M, A, T, and H are all on the circle. A H T

3. What is a circumscribed polygon? • Def: A circumscribed polygon is a polygon with all sides tangent to the circle. P A PAGE is a polygon circumscribed about the circle because , , , and are all tangent to the circle. E G

4. Inscribed quadrilaterals • The opposite angles of an inscribed quadrilateral are supplementary. O NOTE is an inscribed quadrilateral so the opposite angles are supplementary. So ∠N + ∠T = 180˚ and ∠O + ∠E = 180˚ N T E

5. Example 1 Given: m∠G = 62˚, m∠I = 74˚ Find: m∠V G I E V

6. Example 1 continued Given: m∠G = 62˚, m∠I = 74˚ Find: m∠E G I E V

7. Example 1 continued Given: m∠G = 62˚, m∠I = 74˚ Find: m G I E V

8. A parallelogram inscribed in a circle. • If a parallelogram is inscribed in a circle, then it is a rectangle. If OPEN is a parallelogram then it is also a rectangle. Because OPEN is an inscribed quadrilateral the opposite angles are supplementary. In a parallelogram the opposite angles are congruent. Therefore all angles are right angles. P O N E

9. Example 2 Given: ▱TRUE, TR = 40, RU = 9 Find: radius of ʘO T R O E U

10. Example 3 Given: CHAMP is a regular pentagon Find: m C P H A M

11. Example 3 continued Given: CHAMP is a regular pentagon Find: m C P H A M

12. Example 3 continued Given: CHAMP is a regular pentagon Find: m∠P C P H A M

13. Example 4 Can a parallelogram with an 80˚ angle be inscribed in a circle?

14. Example 5 Given: m∠J = 160 – x, m∠M = x2 Find: m∠J J U O B M

15. Homework p. 489 1-3, 5-7, 9, 10, 15