Answer:
[tex]MN=16.4\ units[/tex]
Step-by-step explanation:
step 1
Find PN
we know that
PN is a diameter
PS is a radius
[tex]PS=10\ units[/tex]
so
[tex]PN=2*PS=2*10=20\ units[/tex]
step 2
Find the measure of arc MP
[tex]arc\ MN+arc\ MP=180\°[/tex]
[tex]arc\ MN=110\°[/tex]
[tex]arc\ MP=180\°-110\°=70\°[/tex]
step 3
Find the measure of angle MNP
we know that
The inscribed angle measures half that of the arc comprising
[tex]<MNP=\frac{1}{2}(arc\ MP)[/tex]
substitute the value
[tex]<MNP=\frac{1}{2}(70\°)=35\°[/tex]
step 4
Find the measure of angle MPN
we know that
The inscribed angle measures half that of the arc comprising
[tex]<MPN=\frac{1}{2}(arc\ MN)[/tex]
substitute the value
[tex]<MNP=\frac{1}{2}(110\°)=55\°[/tex]
step 5
Find the measure of angle NMP
The sum of the internal angles of a triangle must be equal to 180 degrees
[tex]<NMP+55\°+35\°=180\°[/tex]
[tex]<NMP=90\°[/tex]
therefore
The triangle MNP is a right triangle
step 6
Find the length MN
In the right triangle MNP
[tex]sin(<MPN)=\frac{MN}{PN}[/tex]
substitute the values
[tex]sin(55\°)=\frac{MN}{20}[/tex]
[tex]MN=(20)sin(55\°)=16.4\ units[/tex]
