Problem-Solving Strategies for Efficient and Elegant Solutions, Grades 6-12 by Posamentier Alfred S.;Krulik Stephen;

Author:Posamentier, Alfred S.;Krulik, Stephen; [Posamentier, Alfred S.; Krulik, Stephen]
Language: eng
Format: epub
ISBN: 1994183
Publisher: Corwin Press
Published: 2008-03-20T00:00:00+00:00

Problem 6.4

In Figure 6.3, is a chord of the larger of the two concentric circles and is tangent to the smaller circle at T. Find the area of the shaded region between the two circles if AB = 8.

Figure 6.3

Solution

We know that a radius is perpendicular to a tangent at the point of contact (T). Furthermore, a radius perpendicular to a chord divides the chord into two equal segments. Thus, AT = BT = 4. We know that the area of the region between the two circles (the doughnut shape) can be found by obtaining the difference between the areas of the two circles. Thus, the area of this region between the two circles equals πR2 − πr2 = π(R2 − r2). Now, whereas OC = R and OT = r, CT = (R − r) and DT = (R + r) (Figure 6.4).

Using the “product of the segments of the chords” theorem, we obtain

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