Question

question
in the diagram below, find the measure of \\( \overline { q r } \\) in \\( \odot c \\).
answer attempt 1 out of 2
additional solution no solution
\\( q r = \square \\) units submit answer

Explanation:

Step1: Recall the property of chords in a circle

In a circle, if two chords are equidistant from the center, they are equal. Here, we focus on the arc length formula \(s = r\theta\) (where \(\theta\) is in radians). But another approach: in \(\odot C\), the radius \(CS = CQ=7\) (radii of the same circle). The arc \(\widehat{QR}\) and the arc with central - angle \(120^{\circ}\) (in the same circle \(\odot C\)): the length of an arc \(l=\frac{n\pi r}{180}\), where \(n\) is the central - angle measure and \(r\) is the radius.

Step2: Calculate the arc length

Given \(n = 120^{\circ}\) and \(r = 7\). Substitute into the formula \(l=\frac{n\pi r}{180}\).

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Another way: Since the two circles (the one with center \(P\) is a distractor as we are asked about \(\odot C\)). In \(\odot C\), if we consider the full - circle circumference \(C = 2\pi r=2\pi\times7 = 14\pi\). The fraction of the circle for a \(120^{\circ}\) central - angle is \(\frac{120}{360}=\frac{1}{3}\). So the arc length \(l=\frac{1}{3}\times14\pi=\frac{14\pi}{3}\approx14.7\)

Answer:

\(\frac{14\pi}{3}\approx14.7\)