Question

if jg = jf, gd = 13, and arc cd = 136°, find each measure.
ed = type your answer...
cf = type your answer...
arc ed = type your answer...
arc hd = type your answer...
arc ce = type your answer...

Explanation:

Step1: Use the property of congruent chords

Since \(JG = JF\), chords \(ED\) and \(CD\) are congruent. So \(ED=CD = 13\).

Step2: Use the property of perpendicular from the center to a chord

If a line from the center of a circle is perpendicular to a chord, it bisects the chord. So \(CF=\frac{1}{2}CD\). Given \(CD = 13\), then \(CF=\frac{13}{2}=6.5\).

Step3: Find the measure of arc \(ED\)

Since chords \(CD\) and \(ED\) are congruent (because \(JG = JF\)), their corresponding arcs are congruent. So \(m\overset{\frown}{ED}=m\overset{\frown}{CD}=136^{\circ}\).

Step4: Find the measure of arc \(HD\)

The measure of a semicircle is \(180^{\circ}\). If \(m\overset{\frown}{CD}=136^{\circ}\), and \(\angle JFD = 90^{\circ}\), then \(m\overset{\frown}{HD}=\frac{1}{2}(180 - 136)=22^{\circ}\) (because the arc subtended by an angle formed by a radius and a perpendicular to a chord).

Step5: Find the measure of arc \(CE\)

The sum of arcs in a circle is \(360^{\circ}\). We know \(m\overset{\frown}{CD}=m\overset{\frown}{ED} = 136^{\circ}\), and \(m\overset{\frown}{HD}=m\overset{\frown}{GE}=22^{\circ}\) (by symmetry). Then \(m\overset{\frown}{CE}=360-(136 + 136+22 + 22)=44^{\circ}\).

Answer:

\(ED = 13\)
\(CF = 6.5\)
\(arc\ ED=136^{\circ}\)
\(arc\ HD = 22^{\circ}\)
\(arc\ CE=44^{\circ}\)