Answer:
[tex]m\angle KLM=53.13^o[/tex]
Step-by-step explanation:
see the attached figure to better understand the problem
step 1
Find the measure of angle KOM
In the triangle KOM
we have
[tex]KO=MO=r=5\ units[/tex]
[tex]KM=8\ units[/tex]
Applying the law of cosines
[tex]8^2=5^2+5^2-2(5)(5)cos(KOM)[/tex]
[tex]64=50-50cos(KOM)[/tex]
[tex]50cos(KOM)=50-64[/tex]
[tex]50cos(KOM)=-14[/tex]
[tex]cos(KOM)=-14/50[/tex]
[tex]m\angle KOM=cos^{-1}(-14/50)[/tex]
[tex]m\angle KOM=106.26^o[/tex]
step 2
Find the measure of the arc KM
we know that
[tex]arc\ KM=m\angle KOM[/tex] ----> by central angle
we have
[tex]m\angle KOM=106.26^o[/tex]
so
[tex]arc\ KM=106.26^o[/tex]
step 3
Find the measure of angle KLM
we know that
The inscribed angle is half that of the arc comprising
[tex]m\angle KLM=\frac{1}{2}[arc\ KM][/tex]
we have
[tex]arc\ KM=106.26^o[/tex]
substitute
[tex]m\angle KLM=\frac{1}{2}[106.26^o][/tex]
[tex]m\angle KLM=53.13^o[/tex]
