Answer:
[tex]BC=11\ cm[/tex]
Step-by-step explanation:
step 1
Find the measure of the arc DC
we know that
The inscribed angle measures half of the arc comprising
[tex]m\angle DBC=\frac{1}{2}[arc\ DC][/tex]
substitute the values
[tex]60\°=\frac{1}{2}[arc\ DC][/tex]
[tex]120\°=arc\ DC[/tex]
[tex]arc\ DC=120\°[/tex]
step 2
Find the measure of arc BC
we know that
[tex]arc\ DC+arc\ BC=180\°[/tex] ----> because the diameter BD divide the circle into two equal parts
[tex]120\°+arc\ BC=180\°[/tex]
[tex]arc\ BC=180\°-120\°=60\°[/tex]
step 3
Find the measure of angle BDC
we know that
The inscribed angle measures half of the arc comprising
[tex]m\angle BDC=\frac{1}{2}[arc\ BC][/tex]
substitute the values
[tex]m\angle BDC=\frac{1}{2}[60\°][/tex]
[tex]m\angle BDC=30\°[/tex]
therefore
The triangle DBC is a right triangle ---> 60°-30°-90°
step 4
Find the measure of BC
we know that
In the right triangle DBC
[tex]sin(\angle BDC)=BC/BD[/tex]
[tex]BC=(BD)sin(\angle BDC)[/tex]
substitute the values
[tex]BC=(22)sin(30\°)=11\ cm[/tex]
