lesson thirty: ‘round about

LESSON THIRTY:

‘ROUND ABOUT

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• 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.

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• 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?

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• 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.

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• 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 =

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• We then divide both sides be in order to isolate r.

12 = becomes

6 =

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• 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

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• 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.

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• 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.

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• 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.

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• We can also say that the circle is circumscribed around a polygon.

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• You will be required to find and draw an inscribed and circumscribed circle in various polygons.

• How will we do this? Any ideas.

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• Firstly, we must find the center.• In a regular polygon, this is relatively easy.• Simply enough, you draw two or more lines of

symmetry and their point of intersection.

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• 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.

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• When finding the circle that circumscribes the polygon, we use the radius of the polygon as our radius of the circle.

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• For triangles, we must recall some old knowledge.

• We must remember incenters and circumcenters.

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• You will recall that incenters are the intersections of all the angle bisectors of a triangle.

• One is pictured below.

C

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• The circumcenter is the point of concurrence of the perpendicular bisectors of the sides of a triangle.

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• 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.

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• 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.

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• 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.

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• 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.

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• This one isn’t too difficult!• http://

www.mathopenref.com/constinhexagon.html

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• This is a lot of material so we’ll take our time with it!

• Do not panic!