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LESSON THIRTY: ‘ROUND ABOUT

LESSON THIRTY: ‘ROUND ABOUT. INSCRIBED AND CIRCUMSCRIBED FIGURES. We already know a bit about circles. You know how to find the radius. You know how to find the diameter. Using these, we found the circumference and area. INSCRIBED AND CIRCUMSCRIBED FIGURES.

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LESSON THIRTY: ‘ROUND ABOUT

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  1. LESSON THIRTY:‘ROUND ABOUT

  2. INSCRIBED AND CIRCUMSCRIBED FIGURES • We already know a bit about circles. • You know how to find the radius. • You know how to find the diameter. • Using these, we found the circumference and area.

  3. INSCRIBED AND CIRCUMSCRIBED FIGURES • But if I gave you the area or circumference, could you find the radius or diameter? • In essence, could you do what we’ve been doing…backwards?

  4. INSCRIBED AND CIRCUMSCRIBED FIGURES • Doing this is simple algebra. • Up until now, we’ve taken the equations or and plugged in the radius. • Now, depending on what we’re given, we’ll plug in the A or the C.

  5. INSCRIBED AND CIRCUMSCRIBED FIGURES • Let’s say I told you that the circumference of a certain circle was 12. • All you must do is use this value for your circumference. 12 =

  6. INSCRIBED AND CIRCUMSCRIBED FIGURES • We then divide both sides be in order to isolate r. 12 = becomes 6 =

  7. INSCRIBED AND CIRCUMSCRIBED FIGURES • What if I told you that the area of a certain circle was 49. • All you must do is use this value for your area. 49 = • Firstly, we divide both sides by

  8. INSCRIBED AND CIRCUMSCRIBED FIGURES • Once we’ve done that, all we need to do is take the square root of both sides. So this… 49 = becomes 7 = r • Thus, the radius is seven.

  9. INSCRIBED AND CIRCUMSCRIBED FIGURES • Now that we have learned this, we can learn a bit more vocabulary about circles. • For instance, when two circles share a center, we call them concentric circles.

  10. INSCRIBED AND CIRCUMSCRIBED FIGURES • When a figure is perfectly inside a circle as in the figure below, we say that it is inscribed in the circle. • This is to say that each of the figure’s vertices are on the circle.

  11. INSCRIBED AND CIRCUMSCRIBED FIGURES • We can also say that the circle is circumscribed around a polygon.

  12. INSCRIBED AND CIRCUMSCRIBED FIGURES • You will be required to find and draw an inscribed and circumscribed circle in various polygons. • How will we do this? Any ideas.

  13. INSCRIBED AND CIRCUMSCRIBED FIGURES • Firstly, we must find the center. • In a regular polygon, this is relatively easy. • Simply enough, you draw two or morelines of symmetry and their point of intersection.

  14. INSCRIBED AND CIRCUMSCRIBED FIGURES • To draw a circle that is inscribed, our circle must touch all the sides of the polygon. • How do we do this? • Well, we simply take the length on of our apothem and make that the radius of our circle.

  15. INSCRIBED AND CIRCUMSCRIBED FIGURES

  16. INSCRIBED AND CIRCUMSCRIBED FIGURES • When finding the circle that circumscribes the polygon, we use the radius of the polygon as our radius of the circle.

  17. INSCRIBED AND CIRCUMSCRIBED FIGURES • For triangles, we must recall some old knowledge. • We must remember incenters and circumcenters.

  18. INSCRIBED AND CIRCUMSCRIBED FIGURES • You will recall that incentersare the intersections of all the angle bisectors of a triangle. • One is pictured below. C

  19. INSCRIBED AND CIRCUMSCRIBED FIGURES • The circumcenter is the point of concurrence of the perpendicular bisectors of the sides of a triangle.

  20. INSCRIBED AND CIRCUMSCRIBED FIGURES • The circumcenter is equidistant from the vertices of the triangle. • The incenter is equidistant from the sides of the triangle. • You can use these to construct a circle around the triangle.

  21. INSCRIBED AND CIRCUMSCRIBED FIGURES • With the incenter as the circle’s center and the length to the sides as your radius, you can create a circle inscribed in your triangle.

  22. INSCRIBED AND CIRCUMSCRIBED FIGURES • With the circumcenter as the circle’s center and the length to the angles as your radius, you can create a circle circumscribed around your triangle.

  23. INSCRIBED AND CIRCUMSCRIBED FIGURES • The last thing I’ll ask you to know is how to construct a regular hexagon inscribed in a circle…given only a point and radius. • The following video will show you step by step.

  24. INSCRIBED AND CIRCUMSCRIBED FIGURES • This one isn’t too difficult! • http://www.mathopenref.com/constinhexagon.html

  25. INSCRIBED AND CIRCUMSCRIBED FIGURES • This is a lot of material so we’ll take our time with it! • Do not panic!

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