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Chapter 13: Problem 9

Triangle \(A B C\) is contained within a circle with center \(C\). Points \(A\) and \(B\) lie on the circle. If the area of circle \(C\) is \(25 \pi,\) and the measure of angle \(A C B\) is \(60^{\circ},\) which of the following are possible lengths for side \(A B\) of triangle \(A B C ?\) Indicate all such lengths. a. 3 b. 4 c. 5 d. 6 e. 7

Short Answer

Expert verified
The possible length for side \(AB\) of triangle \(ABC\) is 'c. 5' only.

Step by step solution

01

Calculate the radius of the circle

Given the area of the circle \(C\) as 25\pi, use the formula for the area of a circle \(A = \pi r^2\) to solve for the radius \(r\). Since \(25\pi = \pi r^2\), divide both sides by \(\pi\) to get \(r^2 = 25\). Taking the square root of both sides, we find \(r = 5\).
02

Apply the law of cosines to calculate the length of side AB

Using the law of cosines \(c^2 = a^2 + b^2 - 2ab\cos(C)\), substitute the given values: \(a = 5\), \(b = 5\), and \(C = 60^\circ\) (which is \(\frac{\pi}{3}\) in radians). Therefore, \(c^2 = 5^2 + 5^2 - 2*5*5\cos(\frac{\pi}{3}) = 25 + 25 - 50*\frac{1}{2} = 25\), where \(c = AB\). Taking the square root of both sides, we find \(c = 5\). Step 3 will evaluate if the possible sides given (a. 3, b. 4, c. 5, d. 6, e. 7) are equal to the calculated length of side \(AB = 5\).
03

Compare the calculated side length with the given options

Only numbers which are equal to 5 can be the possible lengths of side \(AB\). In this case, the answer is 'c. 5'

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Most popular questions from this chapter

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