Question

6 multiple choice 1 point
in the diagram below, diameter \\( \overline { a b } \\) bisects chord \\( \overline { c d } \\) at point \\( e \\) in circle \\( f \\).
if \\( a e = 2 \\) and \\( f b = 17 \\), then the length of \\( \overline { c e } \\) is
7
8
16
15

Explanation:

Step1: Find the radius of the circle

Since \(FB\) is the radius of the circle \(F\), \(r = FB=17\). Also, \(AF = FB = 17\) (radii of the same circle). Given \(AE = 2\), then \(FE=AF - AE\).
\(FE=17 - 2=15\)

Step2: Use the Pythagorean theorem in right - triangle \(FEC\)

The diameter \(AB\) bisects the chord \(CD\) at \(E\), so \(AB\perp CD\) (a diameter that bisects a chord is perpendicular to the chord). In right - triangle \(FEC\), \(FC\) is the radius (\(r = 17\)) and \(FE = 15\). By the Pythagorean theorem \(a^{2}+b^{2}=c^{2}\), where \(c = FC\), \(a = FE\), and \(b = CE\).
\(CE=\sqrt{FC^{2}-FE^{2}}\)
Substitute \(FC = 17\) and \(FE = 15\) into the formula:
\(CE=\sqrt{17^{2}-15^{2}}=\sqrt{(17 + 15)(17 - 15)}\) (using \(a^{2}-b^{2}=(a + b)(a - b)\))
\(CE=\sqrt{32\times2}=\sqrt{64}=8\)

Answer:

8