NATIONAL COUNCIL OF TEACHERS OF MATHEMATICSMarch 2007A parachute company wants to determine the location for aparachute landing target. The goal is to locate a landing tar-get that is as far away as possible from all trees. One wayto accomplish this task is by using an aerial photographof the landing area. On the photograph the parachutecompany draws the largest possible circle that has no treesin the interior. The actual target for the landing area is thecenter of that circle. Once the company determines thelargest possible circle in the area, it locates the center ofthe circle—by tracing the circle, cutting out the newlytraced circle, and folding intersecting diameters. This pro-cedure is related to the first task of this exploration.Defining a CircleAcircleis the set of all points in a plane that are equidis-tant from a point called the center. Find a circular objectabout the size of a salad plate to trace (do not use a com-pass to draw the circle). Trace and cut out the circle.(Make a few extra circles in case you have to start over.) Acircle is two dimensional, while the paper circle is three di-mensional; technically, it is a disk. For the purposes of thisinvestigation, we will ignore its depth and refer to the diskas a circle.The first task is to find the center of the paper circle. Thecenter of the circle is themidpoint, or middle, of the diame-ter. Thediameteris the line segment that contains the cen-ter of the circle and whose endpoints lie on the circle—thatis, the diameter is the longest distance across the circle.1.Why does folding a circle in half locate the diameter ofthe circle?2.What is the fewest number of folds necessary to deter-mine the center of the circle?circle to any point on the circle; the length of a circle’s ra-dius is half the length of its diameter. Open the circle andidentify one radius by drawing a light line along part of oneof the folds. After drawing the radius, fold up the part of thecircle along that line so that the edge of the circle is at thecenter of the circle, as shown below. This fold is achordof the circle, a line segment connecting any two points onthe circle (the diameter is also a chord). It is also theper-pendicular bisectorof the radius. The perpendicular bi-sector is the line that intersects the midpoint of another line(the radius, in this case) at a right angle.Developing the Area Formula for a CircleYou can relate the formula for the area of a circle to theformula for the area of a parallelogram:A=lw. Beginby dividing a circle into smaller and smaller sections. Thephotograph of Montreal’s Olympic Stadium provides a niceimage of these sections.

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3.How many other ways can you find to fold the circlewhile using the same number of folds used to deter-mine the center?Theradiusof a circle is a segment from the center of the

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Term

Summer

Professor

JIM

Tags

NCTM