Question

Need to find OM and NM given AO= 6371 angle NOE= 16.26 deg and that AO bisects angle NOEIf it helps at all the point O is in the center of the circle and segment OA is the radius

Answer 1

As you can see the segments OM ME and OE form a right triangle. So to find the measure of the segment OM you can use the trigonometric ratio cos (θ):

$\cos (\theta)=\frac{\text{ Adjacent side}}{\text{ Hypotenuse}}$

So, you have:

$\begin{gathered} \cos (MOE\text{)}=\frac{OM}{OE} \\ \cos (8.13\text{\degree)}=\frac{OM}{6371} \end{gathered}$

Angle MOE measures 8.13 ° because segment AO bisects angle NOE.

$\frac{16.26\text{\degree}}{2}=8.13\text{\degree}$

The measure of segment OE is 6371 because it is a radius of the circle just like segment AO.

$\begin{gathered} \cos (8.13\text{\degree)}=\frac{OM}{6371} \\ \text{ Multiply by 6371 on both sides of the equation } \\ \cos (8.13\text{\degree)}\cdot6371=\frac{OM}{6371}\cdot6371 \\ \cos (8.13\text{\degree)}\cdot6371=OM \\ 6306.97=OM \end{gathered}$

Now, to find the measure of segment NM you can use the trigonometric ratio sin (θ):

$\sin (\theta)=\frac{\text{Opposite side}}{\text{ Hypotenuse}}$

Also, the NM and ME segments are equal because the AO segment bisects the NOE angle. So, you have:

$NM=ME$$\begin{gathered} \sin (MOE)=\frac{ME}{OE} \\ \sin (8.13\text{\degree})=\frac{ME}{6371} \\ \text{ Multiply by 6371 on both sides of the equation} \\ \sin (8.13\text{\degree})\cdot6371=\frac{ME}{6371}\cdot6371 \\ \sin (8.13\text{\degree})\cdot6371=ME \\ 900.98=ME \\ \text{ Then} \\ 900.98=NM \end{gathered}$

Therefore, the measurements of the OM and NM segments are:

$\begin{gathered} 6306.97=OM \\ 900.98=NM \end{gathered}$